It does sound like you have a short somewhere. I would try disconnecting the front speakers, and see if the problem persists. Then, I would try disconnecting the rear speakers and reconnecting the front ones. If the problem goes away, then you know which speakers are bad, and causing the problem. But, the short could be in the speaker wiring, and not the speakers themselves. I would also check to make sure that you aren't crossing the speaker wires. Of course, they could all be bad, and the problem might be in the steering wheel controls. But, it is probably easiest to start by troubleshooting the speakers. If the speakers are bad, like you said they are, they could be shorting out the radio. If you keep shorting out your radio, you will ruin it.
If you are able to determine that a certain set of speakers, front or rear, are causing the problem, I would check the wiring. Make sure you have positive going to positive on the head unit. If you have the speaker wires crossed, all over the system, it could be causing the HU to heat up and short out.
If you have everything in phase, and still have the problem, you could try running all new speaker wire. I did this when I was having that problem. It actually helped my amp to run cooler. Having different speaker wire gauges creates resistance and heat.
For example,if you have small gauge with connectors, and then run large gauge to the amp, you could have a lot of resistance between the connectors and the different wire gauges. I eliminated the connectors, and the different wire gauges, and just ran one gauge, directly to my power source(amp in my scenario, but it could happen at the HU). If you have bad connectors causing high resistance, this could be the problem too.
If you aren't running an amp, and you are using the stock speaker wires, then I would check the phase first, and then try disconnecting the front or rear speakers. You would want to pull each speaker out and check what color/stripe is going to what terminal(positive or negative) on the speaker and make sure it matches the HU.
To disconnect a speaker set, you would want to do it at the HU, and not the speaker side. And, then wrap the wires with electrical tape separately on the HU side. You don't need to wrap the speaker side. If you do it at the speaker side, you could really short out your radio. Especially, if you don't wrap the wires separately with electrical tape.