After waiting long for Apple to fix these problems and trying my best to develop and share possible work arounds to the problems (see my earlier posts), I find that Apple's podcast app still manages to lose track of episodes that had been on my phone, even if they had been partially played. Even the scrolling of episodes leaves unpredictable blank spaces in the list where episodes should be.
Enough. I've now given up on trying to get by with this broken app. I don't recall any other Apple software of this quality.
I investigated, obtained and compared multiple alternative apps. I found that only iCatcher! and Downcast had support like Apple's Podcast for using the forward and back buttons to jump by chapters within a podcast episode. Most other apps don't even support podcast chapters.
I found that both iCatcher! and Downcast offer additional playback speeds (including the comforable 1.25 speed). iCatcher! lets you configure the time for the forward and back time buttons. Downcast's time button jumps are fixed, but it offers two buttons each for different forward and back times.
They both add gesture support, but of the two, the gesture support of iCatcher! is far greater and is quite handy. After using it, that tips my preference toward iCatcher!