Welcome to the forum , fe2w !
I'd suggest saving work & $$ by giving your item (the oak , anyways) a light cleaning & a couple of coats of food-grade mineral oil wipe-down (not on any 'plywood/veneer' surfaces though) .
Since you've sorta mentioned that the chair has been altered from it's original , another option would be to remove the existing 'manufactured' rattan & replace the seat with a slab of wood , padding & leather cover ... sorta more durable than the ratt & might make the sorta heavy-framed appearance seem more solid .
There have been chairs like yours reworked with rawhide as well .
Looks like a pretty solid oak armchair , so there's value in that .... not so much a 'real' antique (100yrs old or older) , but hung on to for awhile , it'll be one !
If you are bent on refinishing , there's that whole toxic & messy deal , if ya' wanna ..... sometimes the finish just might be shellac-based , which is easy to solventize/recoat (v/s stripping the whole thing out) .
I'm not endorsing any of the above processes as a 100% cure for your ills , but some ARE dirty & cheap !