Audiobookshelf is what your wanting then. It does exactly that, and even has a mobile client.

I believe podgrab can do this

I use podget, which is a 248 kB bash script. I really like it, and think it will meet your requirements:

• It's designed to be called from cron
• It lets you sort your podcasts into categories
• It automatically organises the downloads into different directories based on these categories
• It's been around since 2005 and is still maintained

From its description:

Podget is a simple podcast aggregator optimized for running as a scheduled background job (i.e. cron). It features support for downloading podcasts from RSS & ATOM XML feeds, for sorting the files into folders & categories, for importing URLs from iTunes PCAST files & OPML lists automatic M3U & ASX playlist creation, and automatic cleanup of old files.

It also features automatic UTF-16 conversion for podcasts hosted on MS Windows servers.

I use Pinchflat, but I'll take Youtube channel feeds instead so it can employ Sponsorblock and cut the commercials, especially for podcasters that use IHR. It then exports an RSS feed for Antennapod to monitor, but I imagine you could just write the episodes to a Navidrome-accessible folder.