Repercussions of choosing a poor audio amplifier
"Obviously these speakers have a much higher impedance than the amplifier is rated for, so could this be why the original amp failed immediately?"

NO it cannot. If it would be a problem then such an amplifier would also fail when you would operate it with no speakers installed. That would be plain silly. Audio amplifiers have a voltage output (low impedance) so only a very low impedance could cause it to fail. But almost all modern amplifiers can handle loads of 4 ohms or higher. Also, most modern amplifiers have some form of overload protection so it would just shut off.

OK, it appears there are four 16 ohm speakers in parallel. This would give a total impdedance of 4 ohms. That would be OK if the speakers are indeed 16 ohms. But if they are bit lower in impedance then the combined impedance will be below 4 ohms and for some amps this can be a problem.

A solution might be to instead of switching the 4 speakers in parallel, to make 2 pairs of 2 in parallel and put those 2 pairs in series. Then the total impedance will be around 8 ohms, that should not give any problems.