Chain getting stuck between large chain ring and front mech when changing into big ring
So what's happening is the chain is not climbing the inner edge of the big ring as the derailleur cage pushes it outboard.

I'd suspect worn chain and chainrings, but I'd work through a systematic front derailleur adjustment process to see if I could get it working before buying new parts. The derailleur may have been rotated, or the cage bent slightly.  Park Tool has great videos for derailleur adjustment with a set of steps to go through, here is the front derailleur one.

You should be able to change chainrings, not the whole crankset. FYI the bottom bracket standard from 15 years ago was BSA threaded shells with cartridge bearings (outboard bearings just arriving at this time I think?) which is still very much in use and available.

If you do end up replacing chainrings and chain you should replace the cassette as well. And for good measure replace the cables and housings so you have a completely refreshed drivetrain that will shift well.