Here I've connected my classic to a machine running Monterey:
See the Manage Storage button top right? Admittedly it didn't show for me until I selected Manually manage... which I wasn't sure was entirely safe to do since I've previously synced this with a different computer. But I'm all for testing things, so I did it, and then disconnected the iPod since I've read elsewhere that was a thing that can preserve the existing data. I could always reload from the other machine if it hadn't worked out, so I'm not necessarily endorsing this manoeuvre unless you also have access to all your content somewhere. As it was there were no podcasts on this iPod so the next thing I did was sync some to it. Again I got a warning about removing all existing content, which I cancelled, but I was still allowed to choose some podcasts to sync, which I did.
Clicking the Manage Storage button, and then Podcasts, shows the episodes that I synced, and would allow me to delete them.
While I can find a way to manually remove existing podcasts from a classic I cannot see a way to add them, nor a way to add locally stored podcast episodes into the Podcasts app's library.
See Recover media from an iPod - Apple Community to reclaim all of your media that is on the classic and nowhere else such that you can afford to switch over to automatic syncing. Failing that you may at least be able to switch to syncing podcasts provided you're prepared to replace all podcasts on the device with a new selection from the computer.
tt2