Stripping paint off wooden telephone nook: Help!!

I’m on day two of attempting my first woodworking(?) project in my 1940s house that I recently purchased. I’m attempting to strip the paint off of this phone nook, and I’m on my third round of Kleanstrip as I post this. I see little glimpses of the actual wood but for the most part, what seems to be wood filler is coming off in gobs.

I don’t know what to do from here as far as reaching a point to where I can start sanding. Whatever this goo / paste material is, is still coming up and I don’t think it’s paint. The paint looks more like peels of rubber? I’m at a loss.

Any advice or direction would be highly appreciated!!