I love Player.fm
They have both an android app with a lot of good features (stream or download, can set how many episodes to download at a time, when to delete old ones, all per podcast and as global defaults, set up your own categories/lists, display order, play order, etc) and an actual web site where you can log in and listen to your stuff as well with synced history, so you can pick up pretty much at the exact spot you paused a podcast on your phone (or vice versa).
The “without being signed in” part of YouTube is now no longer completely true. I tried to watch a video tutorial at work the other day and it wouldn’t play because I wasn’t signed in and so “they couldn’t be sure I wasn’t a bot”. I’m not signing into any personal stuff on my work computer, or wasting time creating a “work” Google account, so I guess YT can no longer be a place where I can get helpful programming info.