Amps shouldn't shut down with any sort of reasonable abuse.....
That said, it is better for the amp to shut down than to damage itself..... and better still if the designer can anticipate and design around the abuse that threatens to damage it. But there are limits.
Fire pumps should keep running no matter what.... on board ships they often don't even have fuses in the power line.......
Amps are a tad less important, and can be forgiven if they shut down if played hard in Brownsville Tx in July on a sunny day with no cover and the amp on top of the stack...... if the amp were capable of withstanding everything, it would cost too much for anyone to buy it except the military.
And class-D is notorious for a history of optimists who believed everything about "no heating" of the output devices....
Ideally, the amp should work, and if you do something beyond its capabilities, it should not try, but just not work until you fix the dumb thing you did. If it can tell the reasons for not working, so much the better.
I don't think class-D has any lock on issues, lots of normal analog amps have issues with various types of abuse. But with any newer technology, early units will have more problems, and some folks will be better at making good units than others.
As for the "should be capable of working at full output continuously", there are a number of analog amps that can't do that, and they are in many cases highly rated units that lots of you use and like......
It all depends on what "full rated output" really is. Some amps can put out more than the mains can supply. You can't expect continuous full power from them, even if you could use it.
And in any case, you can't expect full power to be needed continuously when playing bass. We proved that many years ago when we were questioned about it, about the best that anyone did was 50%, and that was not really musical..... more like serious thrash. The use of limiters and compressors can raise average power, but at the cost of dynamic range and again, musicality.