This was driving me crazy also but I seem to have something working.
I created a "Smart Playlist" that I called "Master Podcast List" and made the rule "Genre" "contains" "Podcast". It seems to be working to show only my unplayed podcasts and by using "Shift" and "Delete" and can delete the podcasts once I have played them.
In the List view of the Master Podcast List (icon at the top right corner) I can see all the podcasts I am still waiting to play. At first they were all shown in the list randomly but I clicked on the header for the "Album by Artist" column and found that it cycles among "Album by Artist", "Album by Artist/Year", and just "Album". The "Album" choice is the same way the LIbrary Podcast list sorts them and that put the podcasts in the order I was used to seeing them in. Now I find that if I click on the "Date Added" column it sorts the podcasts that way but still keeps them in the album groups, which is what I wanted.
I can't find a way to get the list view to insert sections like the Podcast list in Library does but if I click on Artist List on the view icon in my Master Podcast List it does pretty much the same thing.
I did not check the Master Podcast List in the "Include Episodes from Playlists" selection on the Podcasts tab of my Ipad and Iphone Devices, just the playlists I wanted to sync and that is working properly also, the playlists for the Podcasts as I already had them are syncing to both devices as expected.
Now I am into the weekend and won't have any Podcast activity for a few days, will have to see what happens next week when the Podcasts start updating to see if this keeps working.
Another way of seeing only unplayed Podcasts is to click on the "Unplayed" tab in LIbrary Podcasts and in that view I find that I can delete podcasts just by clicking the delete key and that makes the podcast disappear from the unplayed group, although they still appears in the List view. Now that I played with that option I can see what Apple is up to. The List view provides a fairly comprehensive list of all the Podcasts that are available and the Unplayed view shows only those that are unplayed. In the Unplayed view I have the option to delete unwanted Podcasts from the view and from the computer but if I make a mistake and delete one I don't want to I can go right back to the main List view and restore it. I still like my Master Podcast List because it gives me the simple line by line view I am accustomed to but the Unplayed view in the Library accomplishes pretty much the same thing.