Update: My problem was caused by the Background App Refresh setting for podcasts. It had been turned off at some point, so even though I was subscribed to podcasts, they weren't downloading for off-line play. After switching it on, my podcasts are now downloading on their own without intervention, and when I'm in rural areas using CarPlay, my podcasts now work great without any cellular reception. One fine note: I'm not sure how frequently the Podcasts app checks for new episodes in the background. Sometimes there's a podcast that hasn't been downloaded despite it's being released. So opening the Podcasts app manually seems to kick it in the rear and grab everything available. But all in all, I'm very happy to have this issue resolved. I hope it helps someone else!
PS: Apple should create a prompt that displays when you open the Podcasts app if Background App Refresh is turned off for the app. It should alert you that even subscribed podcasts will not download with that switched off.