At your mains voltage and 22V @ 3A output, just before the first tap switches
I was also going to mention worst case pass transistor dissipation will be at maximum current and a voltage just before one of the downwards tap changes.
On the reset problem - I'm not totally convinced that the regulator shutting down was resetting the processor without being able to see more than 100mV of ripple on the 5v rail. The ripple we did see was at 5Hz what is that from? Was there some high frequency oscillation the scope wasn't seeing? On the other hand air flow across the board stopping it and the crazy regulator temperature does point towards it.
On the regulator temperature yes it is crazy, the board ought to have a high temperature warning label if the regulator was intended to run that hot.
On the actual design issue I can see how it happens, the attraction of using a common heat sink, the load requirement being underestimated and creeping upwards. Maybe they decided minimum mains input voltage was too high and upped the transformer winding a bit. Maybe the fan was a bit loud and they turned down the minimum speed. What I don't understand is why it wasn't determined to be a problem and fixed before the went into production. I can't believe no one noticed there was a burn hazard on the boards they were developing then building and testing.
Rigol are competent enough to design a 5v regulator, if not their other products would be riddled with problems. Seems more like a management/quality/ethos failure.