I'm going to probably show my ignorance here as my understanding is that 'fixed bias' amps Do need adjusting on replacing tubes (counterintuitive as it sounds) in order to maintain a fixed output voltage.
Cathode bias (class A) are the ones that 'bias ' themselves'
You are absolutely correct - no sign of ignorance anywhere.
One prerequisite for a power tube to work is that the cathode must be at a higher voltage than the control grid (the one carrying the guitar signal). In a cathode biased amp the cathode is at approx. 10V to 30V positive whereas the grid is at 0V. The tube finds its operating point itself - that's why cathode bias is also referred to as "self bias". All preamp tubes are biased this way, too.
In the fixed bias scenario, the cathode is grounded, i.e. at 0V. In order to work we need to apply a negative voltage to the control grid of the tube (usually in the range of approx. -25V to -55V). This voltage is "fixed" in the sense as it does not vary when a signal is superimposed on it. In most amps you can adjust how much negative voltage is applied to the control grid by a pot - that would then be called "adjustable fixed bias" as opposed to say Mesa Engineering and older Hiwatts which do not have such adjustment option - that would be "non-adjustable fixed bias.
Now why can't/shouldn't you change tubes in a fixed biased amp without checking the bias? The idle current drawn by the new tube could be higher than with the old one and be too high with the negative voltage applied to the control grid. Which means that the tube may suffer an early and untimely death. To avoid that, more negative voltage must be applied to the control grid - that's what the tech does by adjusting the pot in an adjustable fixed biased amp.
Now how do I get by in a non-adjustable fixed biased amp? Either the factory applied so much negative voltage (i.e. bias the amp really cold) that it is unlikely that a tube will ever draw too much idle current (that's what Mesa Engineering does), or you have to change the bias circuitry of the amp by changing a resistor.
BTW: cathode biased amps can be biased too hot and kill tubes prematurely as well (e.g. Vox and Matchless amps are well known for that). If too many tubes were killed or shortlived in such an amp, the necessary bias adjustment is to make the cathode resistor bigger.
Cheers Stephan