Gasses can store tons of energy if you get them up to these pressures so it's a hazard to anyone near the thing. If you have gasses in your high pressure system and something fails it can result in shrapnel. Hydraulics can do something similar with spring forces in the casing of the machine, but it isn't nearly as energetic!
There is of course a question about how exactly you build a mechanical device to implement this cycle. In current devices the material which changed temperature physically moves around a circuit and removes and deposits heat energy at different places in the circuit as it moves. They definitely don't talk much about how a practical device could be constructed.
Extract heat to a radiator, turn off the electromagnet, open the coolant circuit valve. It's called Magnetocaloric effect and they were considering it for in-car AC a few years ago, dunno what came of it.
https://en.wikipedia.org/wiki/Magnetic_refrigeration https://www.sciencedirect.com/topics/chemistry/magnetocalori...
if you hang a tensile load of 1000 Newtons from a 1mm x 1mm rod, the rod is under 1 GPa of stress.
This is all I saw to address the pressures required:
> Our higher operating pressures do not represent a barrier for applications because they can be generated by a small load in a large volume of material via a pressure-transmitting medium, e.g., using a vessel with a neck containing a driving piston, whose small area is compensated by its distance of travel.