If I could skip the Post-Play screen entirely, that would suit me fine. When I'm auto-playing I don't care what I just saw or what's coming next, so having a screen to tell me isn't of any use... TO ME. It's a preference for which there is no setting to set, its not directly allowed for. I understand. I could get around it with a playlist when it's that important. But generally, it is not that important as the overall inconvenience of that many playlists and/or creating them would outweigh the benefit of going that route.
Most often, auto-play is an adequate replacement... but not on android in mobile mode.
I run android 5.0 on an Asus Tablet.
From the Plex/Settings/Experience Screen: Play theme music, Show background art, and Auto Play are all checked.
For shuffled movies, the actual rate of auto-playing the "next" movie is nearer to 75% of the time (with the 10 second delay on the post play screen). It will play 3 to 4 movies and then stop auto-playing, requiring me to tap play to begin the cycle again... though it is as likely to be 2-3 movies as 3-4 until I have to do this again.
With TV shows, I actually did get 1 auto-play between episodes after 3 seasons (22- 26 episodes per season) of show (after I posted above). And about 1 auto-play per season after that. Otherwise, I must tap the play button to continue for each episode.
My server is up to date on a fairly fast Windows 10 machine (even when transcoding to more than one device I see less than a 20% cpu utilization hit). My Internet connection exceeds 10Mbits/1Mbit and I have only 4 devices on the radio portion of the router for the local network with never more than 2 devices (1 wired, 1 wireless) using the services of the Plex server locally at once (no plex-pass, so internet serving is not possible).
Basically, what you describe above is what I'm seeing as well. My confusion, I suppose, is why is it "auto-play" if it does not auto-play. Getting a post play screen with a play button you must tap is not auto-play next, it is display next up with an explicit choice to continue or back out. To me "auto-play next" is an advanced implied choice to play WHATEVER is next without further interaction. Currently it only randomly works like this. Plex player TV apps have no confusing behavior in this matter, if you have it set to auto-play next it does... every time until you back out of it manually or run out of episodes. I suppose if I had a remote for my tablet, I could set the android app to TV mode; but I don't, and I don't believe I should have to even if I did. I've not used my Surface in over a year for Plex playing, but it behaved like a Plex player TV app does when last I checked. My iPod and Windows 8 Phone also behaved this way last I checked... but again, months gone by. I have the player app for just about everything but in the last year or so have had less need for numerous devices so I just don't use them.
Truly, I hope that makes sense. I'm not doing anything unusual. I select a movie library and tap shuffle, or I select a TV show and tap play on the Series Screen (not one of the season sub-screens), and then tap first unwatched. When I do this on any TV device, auto-play works as I expect it to... whether I like the post play screen or not isn't the issue, having to interact on the post-play screen is the issue. On TV apps, it proceeds without me having to interact; on the tablet sometimes it proceeds without input, more often it waits for input. It's not consistent behavior within this specific app or across app platforms.
Ironic timing, but earlier this afternoon, I decided to try something different...
On the Plex/Settings/Experience Screen: I unchecked Play theme music and Show background art. Auto-play is still checked.
Results? I still see the background art for TV shows, I don't get the theme music (but I didn't before either) between episodes BUT! Now it fails to automatically proceed to the next episode on my tablet about the same as movies in a shuffle. (I've only had to actually tap the screen once to play the next episode over 5 episodes.) This is closer to the behavior I expect from auto-play, but the other settings should not be impacting THAT setting in the first place. it still says to me something is just a touch off, but less so than before.
While I'm less "personally" bothered, because the behavior I expect is more often demonstrated, it still isn't as consistent as on other app platforms and the setting's descriptions don't help in setting the app up to exhibit the desired behavior.