I never used the podcasts app before but just gave it a try. I can't seem to reproduce what you're describing. On the top-level pages ("Listen Now" and "Browse"), only the purple play button plays the podcast for me, clicking the purple text (the duration of the podcast) does nothing, and everything else opens the detail view for the podcast. Pictures always open detail views. What version of iOS are you on? I'm on 15.3.1.
I’m also on 15.3.1. The huge “Up Next” menu (the first thing on the first page) is what I’m describing - perhaps you need to subscribe to some podcasts before you see it.
Clicking the purple durations next to the play button always starts the podcasts playing for me.
This is admittedly not a flat vs non-flat design problem, but it is exacerbated by the flat design. If actions which started something playing were outlined or something, it would be so much easier to use.