The effect you are describing is “viral load”; the degree to which a virus is present in the body. This is an indicator of how infectious you are. It is especially important for people with HIV to see if they are “safe” or need their medication adjusted.
However, an at-home test will not be a good indicator of this. These have too many variables such as the site that was swabbed, time delays from the various biological functions, how well you used the kit and even variability in the kit itself.
To properly test for viral load, a blood test should be used. I worked with a company that tested for viral load via expelled breath, and while this was a good indicator of infectiousness y/n, and was faster than a PCR, it was not more accurate.
I mean, for 10 bucks anything is a decent deal. Those specs are pretty decent for a simple home server. I’m not familiar with HP thin clients, but I assume you can install a Disdro of your choice on it? My big reason to avoid HP is their crap software and warranties, both of which are moot here.
I would say relatively light software like tailscale, pihole and such would be fine. Docker containers might be pushing it, but that depends largely on what containers you want to run, same goes for nginx; by itself the requirements are fairly low, it depends on what you want to run on it.
Jellyfin might be a stretch, and as you alluded to, real-time transcoding is probably out. It strongly depends on the decoding capabilities of that chip and wether it does hardware decoding or if it all happens in software. The latter might be too much for it. If it can handle it though, it might be interesting as a media player hooked up to a TV, rather than acting as a transcoding or DLNA-esque server.