How do I sync Podcasts from macOS Ventura's Podcasts app to devices instead of old iTunes podcasts?

I've recently moved my media from a PC running iTunes to an iMac on MacOS Ventura - this of course now has separate apps for Music, Podcasts and TV, in place of iTunes.

In the main everything is running well, however I'm not able to sync Podcasts from the iMac to any device (iPod or iPhone) as the option shows the wrong podcasts. As part of the move over from Windows, I cleaned out pretty much all of my previous podcasts and started again with the Podcasts app - the app itself is working successfully, but any attempt to sync to a device is still listing the podcasts from the iTunes environment instead of the ones in the Podcast app.

I think there's probably a setting somewhere that needs to be cleared to have the sync option look in the right place, but I cannot find it anywhere - I've deleted anything that mentions iTunes from the caches, but I still get the same list of podcasts displayed. Does anyone know where I need to be looking for this?

[Re-Titled by Moderator]

Original Title: Device sync on Ventura tries to sync old iTunes podcasts that no longer exist rather than those in the Podcasts app.