I have been listening to podcasts since 2003/4. I have always deleted them afterwards. I already have enough media to handle, adding podcasts to that mix would be a disaster.
But just pushing them to Dropbox/Box/Google Drive/etc is decent enough to do. The nature of podcasts are that you listen, then delete. Even iTunes has it setup to manage it that way when you sync devices loaded with them.
What were you using before BTW?
As for VLC, do you have the Nexus 7 with the Dolby audio?
I've been listening to podcasts around the same length of time. In 2005 Apple fixed EVERYTHING with iTunes/iPod and then iOS integration, but with everything else I'm stuck manually copying stuff over and keeping track of what I'm listening to, etc., just like the bad old days. That's actually THE reason I bought a 5s instead of a drastically cheaper phone, only then iOS 7 podcast support is broken...though not as broken as on everything else. For now I'll just continue using an iPod classic + phone, for as long as I can, I guess...
Is the Nexus 7 with Dolby audio last year's version? The second gen one with a Qualcom chip? I've got the first gen Tegra 3 model.
That MX Player makes me a bit uncomfortable in that I don't know who the heck does it, but it's SOOOOOOOOOOO much better than the Android media player, and doesn't seem to have the audio issues that VLC has on my Nexus 7, sooooo guess I'll use it!
I'll have to play with it more with headphones to be sure the audio's okay, but it all seemed good. Seemed like MX Player "cheats" a bit somehow on both audio and video, making stuff brighter and louder than it would otherwise be, which isn't a bad cheat to have!