I'm gonna say that it's something with the factory amp......either a connection or the amp itself. That'd be my first step. I'd go to a door speaker, or the dash if you have speakers up there....pull the speaker out, unhook the plug. Take a multi-meter and connect it to the plug and turn the stereo on. Let it play at volume until it quits and watch the multi-meter. The voltage will fluctuate with the music...if it zeros out when the thing quits....it's almost positively your amp (99.8%).......since you already replaced the deck.
The other 0.2% is a nightmare.....some stereos will cut out when there's a ground fault (positive wire grounds out).....it could be in the wiring or the speakers....since you have blown speakers, that's where I'd start looking for that. Make sure that the speaker leads aren't touching each other if the cone is damaged......if they're all ok.....start chasing wires.
That's where I'd stop tho....
CHASING WIRES IS HORRIBLE!
....if it's not your amp, I'd assume it's wires and just re-wire the truck....replace the factory deck with aftermarket and run all new speaker leads to the speakers in the truck, skipping the factory amp.
Good luck and I hope this helps.