Ideally in a design everything should be decoupled sufficiently so as to not apply very high frequency stuff to the output of the power supply or back up the supply rails. In theory the power rails should be as solid as a brick outhouse and never allow signals to travel up and down them - at all frequencies.
One other possibility is that many old tube-mic power supplies had many stages of filtering and no 'amplification' where a solid-state regulator typically does the opposite. That solid-state regulator will put out some amount of very high frequency noise, and if you don't do something to get rid of that high frequency stuff, it'll get into your audio circuits too. Even a 'linear' regulator like an LM7815 or LM317 will generate a fair bit of high-frequency hash and hiss. If you want a noise generator, take the noise from a zener diode and amplify it. Funny that voltage regulators do exactly that... Anyways, in this case, proper filtering of the solid-state regulator should get rid of the noise.
Look carefully at the PSRR vs. frequency of an op-amp... and look at the underside of a modern 'Pentium' or similar class motherboard. There's a lot of capacitor there, and for good reason. The instantaneous spikes from switching the transistors in such a processor can be in the 20 to 100 amp region! Enough to upset the power supply rails.
My Soundcraft 2400 has horrible decoupling everywhere. Not enough of it, and some stages even oscillate mildly (only a few millivolts at 6 MHz). Some attention to that and the oscillation goes away - and it sounds much cleaner. The power supply should not make a significant difference on a console's crosstalk unless the channel strip filtering is not sufficient.