Nonsense. The original Mini had the same hardware as the iPad 2: A5, 512kb RAM. The Air, and the Mini 2, stepped that up to an A7 with 1GB RAM. I've owned them all and the iPad 2 and original Mini were not better, much less "a million times better" when running the same version of iOS.
"Million times better" is a hyperbole, but I suppose if it had like you said "512KB" RAM, then it would be much worse.
We also have to take into account that the original iPad Mini did not have a retina screen, nor did it have a 64bit SOC. Both of which significantly increase the RAM requirements, meaning that the iPad Air with it's paltry 1GB was just running into memory issues all the time. A quick search on these forums will show you that, you wouldn't be able to count the amount of iOS System Logs people have posted that contain 'out of memory' exceptions.
I'm not talking about, say, graphics performance with games, where a game can load up eat the RAM and you don't notice. I'm talking about the inability to quickly flick back and forth between apps (copying and pasting, sending messages etc) without having to start from the beginning with them, where you can have several tabs open without them reloading every time you switch between them, or where you don't just get random app crashes because your iPad has come to the end of it's memory.
Yes on paper the A7 will "eat" the A5, just like Samsung phones "eat" iPhones, yet that's not really reflective of the real-world situation. One thing I will say, is that I cannot recall is what the highest iOS I ran on the original iPad Mini was. I believe it was iOS 7 (it had the Ive UI), but on iPad Air is was definitely iOS 7 then 8... Even giving time for optimisation though updates, it just got worse and worse.
My sister still uses that very same original iPad Mini, and it still works fine, despite its age. Certainly still runs better than my iPad Air ever did. My iPad Mini 4 is an absolute charm however! I think I definitely prefer the iPad Mini size anyway... Perhaps my mal experience with the iPad Air was a sign?