Multiple Gateway/Modem Reboots All Day Long

Question

The gateway/modem is dropping phone calls and internet connection anywhere from 10-30 times a day.  The modem is completely rebooting during these issues (meaning it's not just loss of signal ---the entire reboot/restart process happens).
It's totally at random.  We can go up to six or seven hours with no interruptions OR we can get 15 reboots over one hour.  It's at any time of day or night.  Totally intermittent.
That means --- no help.  No techs on the phone or at the house will escalate the problem because they aren't experiencing the issue for themselves.  The techs who HAVE seen the issue replace the modem to fix it because of some reason and then leave...and the issue happens 30 minutes later.
At this point, we have had 5 techs out and 1 supervisor.  
The signal is wonderful (and indeed when the modem isn't resetting, we get excellent speed and connection).  The television and on-demand do not go out.  This loss of connection and reset only affects the internet and phone as far as we can tell.
The wiring in the house is now new.  There is 1 splitter which is now also new.  It's also been bypassed to see if the issue fixes, and it does not.  Random things in the box on the house have all been replaced.  The line from the pole to our house is new.  Meaning ---all these techs and supervisor have "verified" there is no issue from the pole to our modem.
I'm not sure why a line tech can't be sent out if it's "obviously" (their words) not an issue from the pole to our modem.  It doesn't leave a lot of options other than the line to our pole unless I'm missing something.  
Then again, today's tech showed me on his tablet some page that showed all the reboots from our modem just for today.  So, how's it working if he could see it's obviously not?
Two modem replacements, rate codes fixed, blah blah blah.  
Just hand them an incredibly stupid amount of money for no service or fixes I guess?  I mean, I'm dealing with this because I WANT comcast.  I don't want dish or direct.  Google Fiber is being installed in my city, and I'd switch for that.....but I'm the only one fighting to keep comcast.  I love Comcast, but I'm not going to pay for service I'm not getting.  The $40 refund for the last 3 weeks isn't going to cut it when it's STILL not fixed.
The last tech left and 5 minutes later...internet is down and phone is out...and I'm waiting on the reboot.  He switched to a different port on the pole and said he hoped it fixed it because he saw no issues.
So basically...this must be the service available to my house if I want comcast.
Nobody will even look at the error logs on the modem.  One guy who said nothing was wrong with my signal or service said "yeah we see that before we even get to your house" and wouldn't look.  If he saw it, how is there nothing wrong?
My gateway/modem info ----
Cable Modem HW Version: 5Arris Interactive, L.L.C.Boot Version 1.2.1.62Core Version 2.0.5.38Model TG862GProduct 00000000 2AAC4D52Flash 32MBDownload Version 7.6.116.CT DownstreamSNRPower Level37.64 dB37.94 dB37.94 dB37.36 dB38.26 dB37.94 dB37.09 dB36.84 dB-2.15 dBmV-1.83 dBmV-2.00 dBmV-2.19 dBmV-2.32 dBmV-2.94 dBmV-3.76 dBmV-4.61 dBmV UpstreamPower Level 43.50 dBmV44.25 dBmV44.50 dBmV---
These numbers are at the hottest part of the day (since heat can fluctuate the numbers)
Now, the system logs from the 1st replacement modem -- I won't post the whole log, but this sample is an example of what happened through the day.  It was repeat of the same stuff over and over like the following:
Thu Jul 02 13:10:19 2015 Notice (6) Honoring MDD; IP provisioning mode = IPv6
Thu Jul 02 13:09:59 2015 Critical (3) Received Response to Broadcast Maintenance Request, But no Unicast Maintenance opportunities received - T4 time out;CM-MAC=[edit out but available if required]1;CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Thu Jul 02 13:07:49 2015 Critical (3) Started Unicast Maintenance Ranging - No Response received - T3 time-out;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Wed Jul 01 14:54:31 2015 Warning (5) Dynamic Range Window violation
Wed Jul 01 14:27:32 2015 Critical (3) Started Unicast Maintenance Ranging - No Response received - T3 time-out;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Wed Jul 01 11:36:57 2015 Critical (3) 16 consecutive T3 timeouts while trying to range on upstream channel 1;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Wed Jul 01 11:36:57 2015 Critical (3) Unicast Maintenance Ranging attempted - No response - Retries exhausted;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Wed Jul 01 11:36:57 2015 Critical (3) Started Unicast Maintenance Ranging - No Response received - T3 time-out;CM-MAC=[edit out but available if required]:41;CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Wed Jul 01 11:29:40 2015 Warning (5) Dynamic Range Window violation
Tue Jun 30 22:20:39 2015 Critical (3) Started Unicast Maintenance Ranging - No Response received - T3 time-out;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Tue Jun 30 17:06:02 2015 Notice (6) TLV-11 - unrecognized OID;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
Tue Jun 30 16:47:23 2015 Notice (6) TLV-11 - unrecognized OID;CM-MAC=[edit out but available if required];CMTS-MAC=[edit out but available if required];CM-QOS=1.1;CM-VER=3.0;
2nd Modem replacement stopped the T3 and T4 timeouts.  The following is an example of the log both before and after the rate codes were changed.  This sample happens all day long over and over and over.  This is a sample from about 30 minutes this morning.
udhcpd: DHCP server started, 08:51:55, 2015-07-14[System] Logsudhcpd: DHCP server stopped, 08:51:51, 2015-07-14[System] Logsdhcp6: DHCPv6 server started, 08:51:24, 2015-07-14[System] Logsdhcp6: DHCPv6 server stopped, 08:51:23, 2015-07-14[System] Logsdhcp6: DHCPV6 client started, 08:51:19, 2015-07-14[System] Logsdhcp6: DHCPv6 server stopped, 08:51:14, 2015-07-14[System] Logsudhcpd: DHCP server received REQUEST from [edit out but available if required], 08:50:51, 2015-07-14[System] Logsudhcpd: DHCP server started, 08:41:06, 2015-07-14[System] Logsudhcpd: DHCP server stopped, 08:41:02, 2015-07-14[System] Logsdhcp6: DHCPv6 server started, 08:40:35, 2015-07-14[System] Logsdhcp6: DHCPv6 server stopped, 08:40:34, 2015-07-14[System] Logsdhcp6: DHCPV6 client started, 08:40:31, 2015-07-14[System] Logsdhcp6: DHCPv6 server stopped, 08:40:27, 2015-07-14[System] Logsudhcpd: DHCP server started, 08:29:09, 2015-07-14[System] Logsudhcpd: DHCP server stopped, 08:29:03, 2015-07-14[System] Logs 
and from Event LogReceived Response to Broadcast Maintenance Request, But no Unicast Maintenance opportunities received - T4 time out, 08:27:11, 2015-07-14Event LogsLost MDD Timeout, 08:23:02, 2015-07-14Event LogsSYNC Timing Synchronization failure - Loss of Sync, 08:22:58, 2015-07-14Event LogsRCS Partial Service, 08:22:57, 2015-07-14Event LogsLost MDD Timeout, 08:17:22, 2015-07-14Event LogsSYNC Timing Synchronization failure - Loss of Sync, 08:17:18, 2015-07-14Event LogsRCS Partial Service, 08:17:17, 2015-07-14Event LogsNo Ranging Response received - T3 time-out, 07:39:57, 2015-07-14Event Logs
So as you can see, they tell me everything is fine...but if they'd take 3 seconds to look at this, maybe it would help.  If they really "already look at that before" they come into the house, then why keep telling me there is nothing wrong?
I know I'm not getting connection.  I'm not technologically stupid (I'm no expert, but I know when my service stops working how it's worked for years.  I can see big flashing lights as the modem reboots.)
I'm seriously at a loss at what I need to say to these people that I haven't already.  I know it's hot outside, but dang.  It's got to be costing them way more money to have tech after tech sit in my driveway for 20 minutes and then come tell me the service is fine as far as they can see.
In all honesty, I know the techs do what they can.  They've all been polite, and I know the intermittent issue makes it ridiculously hard to figure out.  The thing is, I've been dealing with this issue for way longer than I should be ---while paying a full bill with another due soon.  I'm not asking too much.  I just want service fixed so I can keep giving them my money.  I just need to know what I can tell them other than "just disconnect my service then."
The other thing that points to line issues is that a house on the line before us was torn down.  They jerked some lines off the pole, and they loosened up the comcast line.  The first tech fixed this, but said it might need to be fixed by line techs if service went in/out during wind or rain.  The other techs ignored me, but the tech before today went up and took down the extra wires and all that.  He said he saw no actual issues there other than the amp being "about 20 years old" and some type of connection on the cable about a foot down the line that is no longer used by Comcast.  He said the amp was really old but functional.  Again, none of them want to listen to the glaringly obvious signs that the line techs need to come out and replace some things to fix this.  There is only 1 other house after me on this connection...so if here are issues, there isn't enough crying to comcast to fix it like they would if it affected 20 houses in an area.  

Answer 1

Thanks again, EG.  From the email, I was able to get escalated yesterday.  I was put in contact with a supervisor on 1-800-COMCAST's wireless gateway department. He actually took a few minutes to listen to the error logs I had through all the modems over the last month (over 25 total pages).  He went in and found the dhcp reset error was a common issue with the gateway we had. The gateway is what a lot of people have, and he said it was close to being on the "end-of-life" list soon. So the guys working on our account went ahead and come today and double checked some things. He replaced a switch outside just in case, but he also installed a new wireless gateway that's one of the newer ones they are installing. It's an XB3 gateway. The guys on the phone were great, and today's tech was amazing!  He stayed until the issue was FIXED.  Thank you! The corporate service rep also called back to make sure everything was good, and she is reviewing the account to adjust charges.  The supervisor is also going to monitor the line for 24-48 hours to check for any other issues. The bottom line is that I've tried for 2 ½ weeks to get a tech to actually look at the log, and that was the key to fixing the issue.  So, just updating for the next person who sees the errors I originally posted!  Make them listen to ya!