Stumped! Something in my system overloads amps

Yes, speakers - and their connections - absolutely can ruin an amp. I have a dead Cyrus One in the attic as evidence of what happens when the folly of youth thinks it's a good idea to connect up the speakers with the amp on, accidentally shorting out the speaker leads in the process. (Spoiler alert: it's not a good idea)

There are even some amplifiers (certainly not the ones that Graham makes!) which can be damaged if the speaker connection is lost - not even shorted out - whilst playing a signal. Then you get some amp designs which can become unstable with high capacitance on the selected speaker cables, damaging the amp and/or the tweeters.

There are also some speaker designs (I am thinking of Electrostats here, notably the Quad ESL63 and later iterations) whose overdriving protection circuit places a low-impedance crowbar across the speaker terminals. I think this is supposed to send the amp into protection and/or blow its fuses. You can guess what happens when the amp doesn't have either. But when you consider the hassle of transport and costs of repairing the amp, versus those of the speaker, the crowbar idea makes quite a lot of sense.