I use Apple Podcasts every day and experience none of these problems. Maybe give it another try if and when Spotify’s extra layer on ad logos and banners wears on you.
I was listening to a history podcast (British History podcast), which has been going on for years. I listen while running, so it’s managed on my phone. For a while, I would occasionally get the app trying to skip ahead, but only if it had issues downloading the next episode (again, I was out running).
I eventually just downloaded batches of 10-20 episodes and everything worked perfectly in chronological order. And more recently, it all “just worked” even while new episodes were released. My guess is that some behavior recently changed, but I can’t confirm it.
So, I think the UX goal is to try to get you some content of it has issues with the next chronological episode.