Advice needed for 4TB Time Machine Backups

Question

I run a recording studio based around Logic Pro.
I typically have about 7-8 drives online at any given time with around 5TB, but would like to be able to keep more of it active longer. Most of these drives are backed up by Time Machine. But even with 4TB drives I find I quickly run out of space, and have to either delete old TM backups or buy new drives to keep a running TM archive.
While I periodically archive my stuff to optical, and have Backblaze and Dropbox for server space and insurance purposes, neither of those really address the immediacy and convenience of Time Machine for archival retrieval over the course of months or years. And it doesn't help really with boot drive history that doesn't make it to optical archives and isn't kept on backblaze anyway.
So, I have been considering some kind of Raid device, not necessarily for speed or NAS capability (although those might be nice), but mainly to break the 4TB limit and be able to keep my Time Machine growing as a resource for recalling older projects, etc.
It is likely that this will only be used for TM, although I could see potentially partitioning a bit for something else, depending on how big it is. Not sure.
Things which are important:
•Ease of use. Hook up a FW, USB, ethernet, etc. cable, format some drives, and go. I'm not an IT tech.
•$500 or less is preferred
•4 bays would give me room to expand to greater storage, which I think is the only thing that makes sense. might not be enough additional space to justify bothering with a 2-bay, since reliability is at a premium, I'll probably use the most data redundant RAID settings.
•Reliable -- goes without saying
•Hot Swap and Ability to use mixed types of drives - It's not a big deal to keep a spare 3-4TB WD Red on hand, but it would be nice to be able to use whatever I might have handy lying around in case of a failure.
I have heard mixed reviews of Drobo, and better reviews of Synology products, but again, I'm not sure that I really need a NAS--just something simple and reliable--or as simple as something like this can be.
I welcome any advice on what products and brands people prefer for an application like this. Thank you.

Answer 1

Thanks everyone.
These are some good directions to follow.
The issue I think is defining where in the spectum of backup to archival my needs fit.
The thing is, as soon as you go to an array of drives, both the risk and the cost goes up. To my mind, Time Machine is kind of a middle ground between archival and backup. I can make a manual backup quite easily, but TM automates this process elegantly, and gives you date-specific archival up to the point where storage has to recycle. I can also create fixed archives like optical backups of projects and boot clones, which I do every few months. I see those as true archives. But as a result I've accumulated some 3 dozen cheap green HDDs. Many of them archived TM drives. This is a nice safety feature, but it also gets expensive. Being able to go out to double-digit TB in theory with data redundant raids out to 10TB or more by my estimate would give me 2-3 years of instantaneous TM backup. And I've never been able to get more than about 3-4 months of erffective TM for a while now. Meaning I'm buying a new 4TB drive every few weeks to keep up with this.
Maybe I'm just lazy, but having all of my main drives fed into one TM backup is awfully convenient. I don't have to think too much about backups in the heat of my crazy deadlines, libraries and project data constantly changing and being updated, etc.
A Jbod device and a virtual raid as Grant suggested would fill the bill to solve the problem but it makes me nervous, because the possibility of drive failure will be multiplied by the number of drives in use.
What doesn't appeal to me about any multi-drive scenario is that it's putting all the eggs in one basket and making the TM vulnuerable to corruption or being fried because if it gets hosed, I won't have any archived versions other than the array and my sporadic manual archives. I won't be able to go find the single TM archive on the shelf and pop it in anymore to go backwards. Once I become dependent on multiple drive arrays, there is no easy way to rotate backups, and I will have to do other forms of archival more frequently. It's all a risk-benefit equation and there is no such thing as a free lunch, as we know.
The Drobo 5D is beginning to appeal to me because it at least has dual drive failure redundancy and can rebuild even with the loss of 2 drives. I just wish it had more universal approval in the reviews department. Some people love is, some hate it. I would have to really commit to this thing, and fill it to capacity to amortize its cost over a long period of time.
Thanks again for the responses.