Sample libraries are usually WAVs or (non-MPC) encoded audio files. Mac Minis simply don't have the processing power to handle the DAWs, which are generally CPU-intensive and can be memory-intensive depending on the VSTs or plugins used.
But M1 designs that max out at 16GB don't have the memory to handle plenty of sample libraries, so I don't understand how a Mac Mini is supposed to be up to the job.
It's not just about raw cycles but about cached access to the samples. The biggest libraries can run up to 1TB and you'll probably have more than one. Obviously you don't keep everything in RAM at the same time, but even so - 16GB is a serious limitation for this kind of work.
And if you're using a computer instead of a synth rig you cannot afford to have problems, because any stuttering or glitching is painfully obvious and distracting in a live setting.
It also makes no business sense for a Broadway show that may be grossing $25m a year with a multi-year run to cut costs to the bone on its musical hardware. Considering the cost saving involved in replacing real players (for better or worse...) it makes far more sense to spend twice as much initially for a no-risk professional setup than to pinch pennies and risk glitches.
Memory does pose a bottleneck for huge arrangements in the studio, but in the live setting you literally don't have enough performers at the keys for the same constraint to apply. The stuff they might trigger can be bounced out into multisamples, so the remaining bottleneck is with effects processing.
The new Apple Silicon chips might be a lot faster. But I am pretty sure that audio plugins take a long time to update.
I develop on a VERY old machine in order to support backward compatibility way way farther back than Apple will allow: my current plugins will run on PPC machines because those can be used as music DAWs. As such, the machine I'm compiling on is not producing 64-bit AUs that will work, directly, on MI Macs. They work on literally everything up to that point, but Apple finally shanked me, at least w.r.t that compile target. Until then I was able to support PPC to present day with one three-target fat binary :)
Another dev, Sean Costello, told me that older builds of his stuff (pre-2017?) weren't running on M1, but everything built past a certain point (a new version of XCode, that had long abandoned things like PPC and possibly 32-bit support) was automatically working on M1 through the Rosetta layer.
So, depending on the build environment, Apple arranged that the audio plugins don't even have to be updated. Depending on the libraries the plugins rely on (a vulnerability for some of the big names that use bespoke but OLD libraries to do things), some of the plugins might need only a recompile to be native to M1 architecture. And some might be really intractable.
Also, if you need to replace or duplicate a unit with an identical one, you know you can always run out and easily find replacement Mac Minis. If you used some random ultra-SFF PC, can you be sure you can get another identical one easily if you need to? What about 3 years from now? Changing out hardware always introduces a possibility that something might go wrong - exactly what you don't want on a broadway show, a few hours before a performance!