I've never powered my radio down before I shut the engine off. No issues. That should be a non issue. Just like turning your AC off before powering down. It hurts nothing for your vehicle to power on with the AC turned on. Back in the day, it used to put extra drain on the compressor to do that, but these compressors today don't have an issue with that. Same with radio. It should be a non issue having it set to on when you start the vehicle. I would take it to a Dealer, or just call them, and see if there are any active recalls on it. Outside of that, someone will need to diag it and find out what's up.