Two less than ideal work arounds for now ...
1) Use the method originally suggested of removing the Podcasts app via iTunes. Then ... when you want to load new podcasts, you sync the app back on the phone, then sync your new podcasts ... then remove Pocasts app again and restart phone. (I know ... very tiresome!)
2) If your JVC stereo has a USB connection there's a decent chance it will have an auxillary small stereo jack connection. Then you just need a male to male stereo jack cable. The downside ... you have to control the podcast play with the iPhone itself. But this should be easier to do using the Podcasts app.
I have a work around to get the Podcast App to work with my JVC USB connection, but it takes some work each time you start the car. Note: This is for the IPod 4G (6.0). I have my JVC settings to "IPod Mode" (controllable from the IPod).
When the radio is on and the IPod not plugged in, open the music app, then plug in your IPod to the USB. It should now find your music and start playing music. Pause the music, and press the back arrow (upper left corner of the screen), then press the Home Button. Open the Podcast app, then open the podcast you want to listen to. The podcast should start playing through you JVC USB connection.
I suggest you do not try to drive during this operation. You need to do it each time you restart your car, it will not start playing automatically. I have other Apps like Tunein Radio that work correctly, so this looks like another Podcast bug that should be fixed.