It can be hard to say if it’s the Apps fault (ie a bug) or we are not yet understanding it and setting things up properly - a lot seems to have changed and I’m still having issues adapting.
This is Apple’s guide Subscribe and listen to podcasts - Apple Support that may help, I’ve also found deleting the App and then re-downloading from the App Store can sort a few things out. The last resort is to Restore your device, a major hassle but again if something isn’t working correct can resolve, even when you bring back your settings via a BackUp - think the process clears caches etc but could be mistaken about that reason.
A small point, when you delete the Podcasts App (from Settings/iDevice Storage) that process leaves alone your Subscriptions, so when reinstalled and opened for the first time they should all come back (and more importantly be a reflection of what’s on the Mac).
Hope the above helps.