Nuncio - OK. One of my 'problem' shows is the BBC's 'Beyond Today' - Stitcher just cannot 'see' it, although RadioPublic can. Interestingly - and this could I suppose be part of the problem - there is another show of the exact same name, published by the 'United Church of God' (which as far as I can tell is not a covert BBC outfit!) which Stitcher CAN see.
Podbean can find BOTH these two shows and so can Overcast, if I just use their Search on their webpage. Since iTunes was much earlier into the whole podcast thing and is very widely used by creators, maybe iOS-based stuff is more likely to work, but since I don't have an iPhone that's no use to me.
It's interesting also that you say "Except for some BBC podcasts, I've never failed..." - another difficult show is the BBC's 'Political Thinking, with Nick Robinson' which Stitcher steadfastly fails to find, although it will get a number of other BBC Radio 4 items. Both Podbean and Overcast do find this.
A further oddity surrounds the Patreon service. I support two or three podcasters who publish ordinary podcasts but who look to garner a bit of financial support by publishing separate, supplemental episodes via the Patreon platform. To get these 'member-only' episodes, one has to get a different RSS from Patreon, because they are in effect a separate podcast. Stitcher would get the main show but not the Patreon supplemental and RadioPublic was vice-versa - bizarre!
Grams - interesting. I hadn't picked up the concept of - if I understand you correctly - 'proper' podcasts vs. 'not podcasts' although I had assumed that the monetisation aspect would likely be a factor in this somewhere. Perhaps the 'walled garden' is opened only to publishers who accept advertising being attached to their stuff (as seems to be the case with Stitcher)? I have sometimes found that an app won't work with the plain RSS feed string but will accept the 'Copied link' from iTunes.
Ben T - yes I too value the 'skip' button - both Stitcher and RadioPublic allow you to adjust the period forward and back independently, which is really useful. All the apps I've tried so far have had 'Playlist' options but I haven't used them yet, since I'm still struggling to find one I like!
Edited to add: I don't mind paying to remove the advertising but I do need the app to meet all the other requirements first.