2005 stock non-nav radio suits my purposes fine except that when on a long drive I like to listen to podcasts on my iPod to help pass the time.

Crutchfield says that an auxiliary iPod integration unit is not available but a wired Scosche FM-MOD02 is. Reviews often state that it is better than a wireless transmitter but still has issues.

I would be willing to replace the stereo but I have read that it is difficult to find another unit that looks good with all of the silver. Not sure if that would be an issue in my car as the previous owner did replace the silver with wood grain.

 

I have used wired FM adapters several times and I am a HUGE fan of them. Basically they intercept the antenna, and bypass any interference that you get with wireless FM adapters. The only issue is they all use standard style antenna jacks, and you will either need to get a couple of adapters for each end, or like I did in one of my older cars, cut your factory antenna coax in the middle and put in a male and a female antenna jack which you will put the wired FM adapter into, and if removed, you just hook them together.
The sound quality is much better than the wireless ones, and they work perfectly. I also thought about doing the same thing, but wanted a more featured radio, but honestly, I may go back and do the same thing you are talking about.
Just hide the switch in the glove box, and run the cable there. I currently have that exact device in my 1967 El Camino so I can listen to audio books from an iPod in the car.