My chair suddenly started lurching precariously from side to side, and when I took a closer look I noticed this inner ring around the cylinder that keeps the chair centered had popped out and vomited a bunch of grease everywhere.
I can push this thing back in and it'll sort of click into place but a few minutes of sitting later the chair starts wobbling again.
It's an unrepairable replaceable part, you need to get a new gas cylinder that fits or a new chair
Or buy it again new, then stick the busted cylinder in the box and say it came like that and get a refund
Kinda looks like a busted piston. Depending on the model you might be able to find a replacement piston online
Is the chair base itself warped or damaged? The place where the cylinder sits?
IF the base itself/the hole that the cylinder sits in IS bent, warped, damaged then unless you're okay replacing both the cylinder and the base at a cost of probably $80-$90 which is getting pretty close to the cost of some budget chairs then you may have to consider whether it's better to just get a new chair.
Usually the base itself is just the 'arms' that hold the wheels plus a central hole that the hydraulic cylinder fits inside snugly (you often have to hammer it to get it out again when replacing but it can be done).
If the base is fine however you can buy a new piston which should fit 90% of office chairs (no idea about gamer chairs) online that comes with a toolkit for doing the replacement for $40 on bezos-net.
If this is not a normal office chair and I'm not sure it is from the picture (it should have levers under it for adjusting height) then I'm less sure.
I'm pretty sure the base is okay. It is, however, a budget chair- a budget gamer chair. I have some old broken office chairs that I've kept around (the backs and arm rests fell off because they were cheap plastic, the seats and bases are fine. I'll check if the pistons are interchangeable