recommendations for a cheap single player first person shooter I can download on Xbox Series X, and jump right into shooting ~~peeps~~ non-sentient enemy constructs ASAP?

DOOM (2016) is almost always on sale these days, and it drops you straight into the action.

Halo master chief collection (don't know if it's cheap), but I think they have firefight mode which sounds exactly like shooting peeps asap

I also like old school single player games often to zone out playing.

Titanfall 2 was good mindless fun, it is like $6NZD on steam, no idea what your currency is.

I bought the Castle wolfenstein reboot games on steam recently in a bundle and that was great fun for the $17NZD.

Also the Halo master chief collection was approximately $25NZD at one point and that was great fun on a dollar per hour basis.

The mass effect games were like $10 at one point too, and the first couple of those are great.

Again, I don't know much about the xbox pricing on these, they might be still gouging on that platform.

Quake 3 Arena is quick fun and is available for free: https://archive.org/details/quake-3-arena

Splitgate my dude

If Borderlands 2 is available in that library, I can assure you it has really held up to the test of time.

One of my favorite games of all time in fact. I have played it so many times I have lost count. If there were something like this that I could play new, that would be amazing.

Titan Fall 2

Got Far Cry 5 basically for this, maybe Crysis too. I've seen Borderlands mentioned here and i agree

Robo quest, Doom 1 2 64, dark forces remaster, and bolt gun were all pretty fun and quick on the shooter action

If you're just looking for a cheap PVP FPS, why not try one of the many free ones like Fortnite?

Single player for me. I can't keep up with these kids. Also hate battle Royale.
- and jump right into shooting peeps
  Your title says PVP