Is there a podcast app that can download podcasts so that they're available to the stock music player
The BeyondPod podcast player will download new podcasts directly to your device, on a schedule. For example, I have it run every morning to look for new podcasts.  And, it can be configured to "Share Podcasts" with the default Android Music Player.

I have never used the share-with-default-player feature, so I can't really comment on how well that works in practice.  But the basic podcast management and download features have all worked nicely for me.