How does the hospital equipment double check that an individual internal crystal timebase has not “gone rogue”?
It's not really an issue. The service life of say a morphine pump is determined by life of the parts that last only a tiny fraction as long as the quartz (typically)crystal. Piezoelectric materials like quartz are used for their accuracy. When you apply current to the crystal, it's bound by the laws of physics. It will oscilate at X frequency as per it's size and cut, and unless an outside force acts upon it, or the steady decay of the crystal (measured in lifetimes) alters it physically, it cannot oscillate at a different frequency.
Another thing not mentioned here is that relying on the hertz from the socket isn't a very accurate measure of time. One of the great boons of the digital advances in the 70's was that quartz time keeping was much more accurate.
Also, note that the quality of materials used for time keeping varies. The ones say in cheap consumer clocks and watches are manufactured by the billions every year, and go for pennies. On the other end of the spectrum some are made to extremely precise specifications, and can be rather expensive. You can bet that the X-ray machine which also uses piezoelectric oscillation to control the length of the burst, isn't using a cheap one. Btw, your nurse is your primary defense against any malfunctioning equipment in the hospital.