If you wanted to remove the fabric and old padding, one thing you could try is to cut away the fabric at the first line of staples. You should then be able to remove the old padding and make repairs to the webbing as needed.

I’m reupholstering my very old dining room chairs (my parents got them from a yard sale when they were first married and I’m 30 just to give you an idea). They have the pop out cushions but with an extra lip on them so it’s not as straight forward as all of the videos I’ve seen online. These seem to require two levels of staples. That being said, the cushions are in bad shape and very uncomfortable. My mom did the first re-upholstery of them way back and used a ton of staples. I’m worried about pulling all those out (time consuming and because the wood seems fragile) but the batting in there is probable corroded. Should I just add a layer of foam to the existing fabric and put the new fabric over that to avoid a disaster and potentially having to buy all new seats? Whatever advice you have would be much appreciated.