I know OP copped to “various Caffeine/Amphetamine apps” but said they're often “overkill”. In this case I think a system-provided command is less overkill than a web page that either uses a browser API or plays an empty video.
I like that OP’s site turns keeps my machine awake while it’s open, and I can just close the tab when I want my machine to behave normally.
If you want. If you don’t, use the `-t` flag for it to turn off after a set number of seconds or `-w` to do so after a specific process ends.
You can tell caffeinate to watch for other processes too, to make sure a browser thread doing a download will finished uninterrupted before sleeping.
I do a `caffeinate -d` and then throw my mouse to a hot corner to activate the screen saver.
I think you buried the lede there...
https://www.ifixit.com/Wiki/What_to_do_with_a_swollen_batter...