Might not need to glue the mat back down. I'd clean off the old residue and try it without the glue. My old Pioneer was like that, but that was so you could get to the belt with the platter installed.
Check out Mr. Carlson's Lab on YouTube:
https://www.youtube.com/channel/UCU9...wrWb_3GxeteQPA
He talks quite a bit about capacitors and other components and has recommendations on high-quality caps. If I recall, he likes Illinois Capacitor for polypropylene capacitors, Rubycon Capacitor for electrolytic.
What else do the speakers need? I'd be ok with doing the crossover and wiring work, but I wouldn't trust my skills on mechanical repairs on the speakers themselves.
I've been wanting to go to the record stores, but can't drag myself out of the house in this heat. I think I'm done trying to get vinyl from eBay - hasn't been a very positive experience overall. Some have been good, but over half have had issues and there are many unscrupulous sellers there. Used vinyl from Amazon has been ok so far. Thrift stores have been a complete bust. Nothing at yard sales so far, but we don't have yard sales very often around here.