If the pin is too small for a standard base, you may be able to get a metal reducer/bearing or something to make the hole in the base narrower. Like this https://www.lowes.com/pd/Hillman-1-Count-1-2-in-Bronze-Machine-Bushing/3018257

A cylinder such as this https://a.co/d/59kZeu3

Base such as this https://a.co/d/i6epzlH

If the new cylinder doesn’t fit the existing chair hole, maybe you could couple the metal rods together with something like this https://www.lowes.com/pd/SteelTek-1-1-4-in-Silver-Galvanized-Steel-Structural-Pipe-Fitting-Coupling/999931022 That piece specifically might not be long enough to securely grasp both pieces, but it gives you an idea what to look for.

That metal shaft sticking out of the bottom might come out of the chair, leaving a hole in that black piece into which a new desk chair base and cylinder could be installed. If so, that shaft might be really wedged into the chair, may take some creativity to dislodge it.