How images come to on the surface of our monitors ...

I'm a newbie in color management and I want to ask a question to a real color guru ... thanks a lot in advance.
When we look at the sRGB monitor, the image we see in ACR, comes to us with the following steps:
1. Linear raw image,
2. Linear RGB data after demosaicing,
3. Linear input profile interprets the RGB data created by demosaicing process and send them to the PCS for step 4,
4. Gamma encoded any color space we choose as a working space (ProPhoto RGB for example),
5. sRGB, at the monitor ... we see these sRGB image on our monitors.
Are all these steps correct?
Or, any missing step?
And, additionally, input profiles are always linear?

Thank a lot Andrew, twenty-one ...
I'm especially intrested in the functions of input profiles in the process.
Let's leave soft proofing aside .... and let's say camera is in JPEG mode ...
How many steps are there in cameras for creating a JPEG image from the color management perspective?
And, also let's leave aside the preview display on the camera LCD.
Sensor gathers linear raw data and demosaicing process creates three channel, linear RGB data ... this linear RGB image is untagged and has very large gamut ... this untagged RGB data are then converted into the specified RGB space (sRGB or Adobe RGB for most of the cameras) by the input profile.
So, the process in cameras for JPEG creation has the following steps: sensor raw data > untagged, very large gamut RGB by demosaicing > input profile converts the untagged, very large gamut RGB to the specified color space (sRGB or AdobeRGB) ... and then we have JPEG file in our hands.
Are these steps correct?
Thanks a lot for your time ...

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