I’m going to buy the brushed aluminium handbrake cable from Jass: https://www.jassperformance.com/shop/mazda-mx5/By%20Model/MK3/handbrake_handle
Has anyone bought this, how well does it fit and function?
Also, more importantly, how does the standard handle come off?
I fitted one from Mx5parts, the original ones just twists and pulls off. I say that but a little more effort is required, mine was b***** tight but got it off eventually. The new one was equally tight to put on.
Ah I thought as much - I’ve tried to twist it off before with no luck, guess I’ll have to try a little harder! 
Probably advantageous with the cold weather to apply a little heat to expand the grip/handle, same with the new one before pushing in on. I found a grippy rubber glove helped and a good deal of cursing comes in handy too.
The push button release twists off too but that’s threaded, normal RH thread.
Right on, I’ll crack the hairdryer out!
So you have the hairdresser tools?
Yep!
I find heat guns can often be too hot and cause damage, a hairdryer brings the temperature up more gradually!
Stolen from my Mrs 
I have done this and concur with MickA, warm it and twist it off. Btw im a MK2 but imagine it’s probably the same.