Make iPhone imitate iPod for podcasts

I doubt you'll be able to do this through the Podcasts app. Caveat: I gave up syncing podcast files (as podcasts) between my computer and iPhone years ago because it just didn't work anymore; maybe it's improved now, although I doubt it. What you want is kind of possible, and it's still what I do, but it's a heck of a lot more complicated than it was seven or eight years ago.

What I do: on the computer, change all downloaded podcasts to Media Kind = Audiobook or Music. (Select a file, Command-I to Get Info, select Options, and then in the Media Kind menu choose Audiobook or Music.) Note: those podcasts will then download a second time - I just mark them as played the second time.

If you've got the files as Music or Audiobooks, then you can sort them into playlists, smart or dumb, and then sync them with your iPhone.

If you change the Media Kind to Music: the playlists behave fairly predictably, but you're going to lose things like jump 15 seconds. The once-podcasts-now-music will play through the Music app on your iPhone. This may also do weird things if you're using Apple Music and iCloud Music Library or whatever it's called. (This problem is one reason I do NOT use Apple Music.) If you've got music playlists that are based on "Media Kind = Music", you're also going to have issues.

If you change the Media Kind to Audiobooks: you retain the brief-jump function, so that's nice, but the once-podcasts-now-audiobooks won't (usually) play through Music - instead, they'll be in the iPhone's default iBooks app, and that leads to a different set of problems. The playlists you set up in iTunes will determine what you can sync, BUT those playlists don't actually appear in the iBooks app. Instead you'll just see all the different content organized by title or author or whatever criteria you select. Additionally, the play counts become totally messed up, at least in my experience. For example, I've often got 10 unplayed episodes of "Stuff You Should Know" synced to my iPhone, and if I use the Audiobooks app to play just the seventh of those episodes, then #1 through #7 seem to all get marked as played (and often marked as played twice, who knows why).

What I'm using instead: Audiobook Browser, an app designed by someone who's active in this forum. It's not the prettiest of apps, but it maintains playlists and unlike in iBooks, in Audiobook Browser you can see the playlists. I don't believe it can play episodes in succession - just one, and then you need to select a different episode. It works much better than the iBooks app for playing synced playlists of short audiobook (originally podcast) files, and it handles play counts predictably.

So my workflow:

1. Podcasts that I listen to and then delete (news, for example): I use Overcast to subscribe to these on my iPhone. I now avoid Apple's Podcasts app and my life is better for it.
2. Podcasts that I want to keep: I subscribe to these on my computer (and every option that even hints at syncing is turned off). Manual refresh only. Every few days I "refresh" and download them, and change the Media Kind for all of them to Audiobooks. Then I refresh again, and the same podcasts download again, and I mark those extra files as played.
3. Every few days I manually sync my iPhone with my computer, and the once-podcasts-now-audiobooks copy over. (Of course, that only happens if my Photos library is cooperating on that particular day, but that's a whole other issue.)

It's kludge upon kludge upon kludge, and it seems like with every iOS update, Apple adds another complication.

Good luck!

I wanted to continue to manage my podcasts in iTunes also, so had to find an effective way to deal with syncing what I wanted onto the device and play them in the Podcasts app also. Here's what I'm doing that works for me:

(1) Create a playlist in iTunes (select the Playlists view under Podcasts and click on the Plus button down the bottom to add a new playlist, then rename it)

(2) Drag whatever podcast episodes you want to sync into that playlist (you can see your full library in the Playlists view by selecting Library > Podcasts in the left panel). Then reorder the playlist however you want.

(3) In iTunes, with your device attached, select the device, then Podcasts at the left. Scroll the page down to see the section Include Episodes from Playlists and make sure your playlist is checked. You can turn off syncing of the podcasts themselves if you just want the episodes in the playlist. Then sync to copy the playlist and included episodes onto the device.

(4) In the Podcasts app, under My Podcasts, there is now an iTunes Playlist item at the top of the list. You may have to scroll to see it. Tap on that, then your playlist, then the first episode to queue up the playlist.

Having done that you can just update that playlist in iTunes and sync in future. You might need to reselect the playlist in the Podcasts app after syncing. That's working for me and saves me from having to manage podcasts on device or flip around choosing podcasts in the Podcasts app- I can just start my playlist going and let it run through and then update & sync that weekly. Doing that, or a variation on it, may suit what you're looking to do also. You don't have to play through the playlist if you don't want to - you can just use that mechanism to be selective about what comes across to the device.