I have similar questions to Doublej as to what "very old" means. It would help to have some specifics. If it's 'PowerPC'-old, maybe even older Intel mac, you may want to run a suitable Linux on it rather than an older MacOS that is unsupported for security updates etc,. I'd also be leery of an original HD on a very old machine. Even though your data is external, you might want to consider a preemptive HD replacement to avoid later problems (they're cheap). Since you dont need a lot of internal storage, maybe even a small-capacity SSD.
You should be able to put together a great music delivery system this way. I'd probably opt for the Mac Mini. It will be the most unobtrusive. You don't need all that much computing power (by today's standards) for what you want to do and you can run it 'headless' (no display) and control everything from a smartphone or tablet.