Yup that's my one issue, otherwise I'd swap in a heartbeat. You can do it through nextcloud servers but not any free one so far as I can see. Wouldn't know where to start with a paid alternative.
Edit: Podverse, also FOSS, has a sync feature as part of their subscription. It's $18/year. Haven't used it to know how well it works, but much cheaper than Pocket Casts anyway (and a 3 month free trial it seems). It's on F-Droid.
Well the sync across devices would be the main thing that people would deem worth having, just not at that price.