I can get podcasts to work with my iPod on my Mac - Catalina 10.15.1, however, it is a pain.
Prior process on iTunes:
1) Plug in ipod so it can update
2) Eject iPod and go.
New process in Catalina:
1) Plug in iPod.
2) Open finder window
3) Select ipod
4) Resize finder window because it is too small to see the podcast items
5) Select podcasts (don't sync yet)
6) Open Podcast app
7) Refresh podcasts so new ones are loaded
8) Return to finder window
9) Select podcast I want to sync
10) Unclick "Sync Podcasts . . ." to make sure all the old ones ares cleared off, it doesn't always remove old ones
11) Click Remove button
12) Click Apply
13) Click Sync
14) Click "Sync Podcasts . . ."
15) Check to ensure that the correct podcasts are showing up in the finder window(if not, that is a whole other process)
16) Fix any issues with the podcast list
17) Click Apply
18) Click eject
Now at this point the real fun begins because if you were in the middle of a podcast, that location is long gone, it goes back to the beginning Also, if making multiple updates (i.e. several podcasts from several days) there is no guarantee that it will play the oldest first. Which means you can either let it play what ever it wants and fix the problems as mentioned above, or hunt around for it. Note that if you just even start a podcast it gets marked as played and you will have to manually reload it each update.
Bottom line: The way Catalina handles iPods is absolutely horrible (not to mention the number of times my Mac has crashed or rebooted itself).