It's highly specific to your setup and the game/software. Most games aren't a problem. Just the occasional random issue, like in WoW certain locations insta-crash my graphics driver.

ancient 80GB HDD, ancient entry-level single-core CPU and 1GiB of RAM

It couldn’t even run Linux Mint XFCE (crashed after opening Firefox)

So the biggest limitation for literally anything will be memory. 1GiB is less than anything other than an Ubuntu server VM will handle

Pro editions of Windows 10 have memory compression which combined with paging will allow it to barely function, but Windows 7 and later will absolutely chug on a single core processor, with 10 basically being unusable due to heavy background processes.

On Linux it appears you have to really do some heavy customization to get memory compression to work, but you can use zram-config to setup a compressed swap file, so it will be slightly less bad. I suspect this is probably the easiest path to having this computer be capable of loading Firefox and a GUI.

With all of that said, an 80GB HDD is going to be incredibly slow even by hard drive standards, and a single core processor is going to be missing so many modern instruction sets that everything will be slow as molasses but even worse, it'll be unreliably slow because certain things that rely on those instructions will chug as it churns through it the hard way, but then other things will zip by normally.

This PC sounds like an excercise in refusing to let the dead die, which while an entertaining challenge, eventually the only solution will be to make it place for running period-correct software