AntennaPod is what I've been using without issue — fully free. Fits your criteria AFAIK, only exception I can think of is that it is based on your phone since you aren't making any account for it to track stuff online, so it'd forget if you changed phones. (Tho some data can be backed up manually.)
Only got as far as putting in rss links (which was unnecessarily difficult because old app didn't have a way to copy them out) but this *feels* like what I've been wanting for years. Thanks again.
Thanks, will look. I only use my phone. Sonnet, which I'm currently using, has started intermittently losing my place if I pause mid-episode for several days. Or if it just feels like it.