They come in all sorts of varieties - plain mince, mince and cheese (yum), steak and cheese, mince and potato, and then some exotic varieties like butter chicken pie.
They're typically cheap and made from bad quality ingredients, but you can get very good quality mince pies if you know where to go.
If you're a hungry teenager and you need a snack - nothing beats a mince pie.
A mince pie is the sweet fruit tart that is essentially only stocked at Christmas time.
Meat pies are all over the place, and are popular enough that a former premier referred to them as Australia's national dish.
The fruity kind of pies that only appear at Christmas time are called "mince pies", but we should really find another name for them because it's a constant source of disappointment when you expect savoury goodness and are presented with a sugary lump.
If it ever contained anything one could in good faith call 'meat' (by modern definition, at least), it's news to me. [Edit below]
(The brandy however is borderline essential! Certainly a mince pie ought to be served wirh brandy butter.)
That said.. I don't think it's relevant how the quirks came to be, as in, it's no less interesting if it's easily explained away as 'oh well that used to make sense', because it's still funny/ambiguous/confusing when you take a step back and take everything at face value.
[Edit: you are absolutely right: https://en.m.wikipedia.org/wiki/Mincemeat - it was essentially preservation by naturally sourced citric acid and sugars. News to me indeed! I stand by above though.]
So maybe this is not good example of English being like a Mark Twain's German schizo language. Gastro-culture changed on it and the folks at Merriem-Webster didn't get the email, that's all.