Was Metallica really responsible ?

He had some classical training in college and some piano lessons, which I think is different than a trained classical musician, especially since he was not playing any classical instruments.

Be that as it may, I don't understand the distinction you are making with "neo-classical" and "incorporating classical music into metal". There is no obvious link between Burton's parts and classical music, except for the quiet part in To Live is to Die, which is way too late to credit the band for accomplishing what you say. What you call "subtle", I would call "essentially non-existent". In fact, I would argue that what you say Blackmore and Purple hinted at is way more prominent than anything Metallica did (like the live concerto - Metallica's S&M is shit by the way, just for the record). Cliff's contribution was knowledge of harmony and such, which he passed on to Hetfield and of course his free-spirited sense of songwriting which greatly influenced Metallica, hence the huge difference between Kill em All and Ride the Lightning. The adventurous songwriting is certainly not something pioneered by Metallica, even though I happen to believe that there are few things more perfect than the song Master of Puppets.

However, the main point is that you are grossly overlooking Malmsteen who truly fused classical music and metal, before Metallica and a gazillion times more notably than Metallica  - and his influence can be traced back to Blackmore partially (but how much further) and his documented studying of classical composers (including Bach and Paganini). His influence is undeniable. The fact that they called his stuff "neoclassical" is the dead giveaway. And every genre of metal that has classical music influences takes its cue from Malmsteen really. Before him of course there were many who had traces of direct classical music influences, like Blackmore and Rainbow (Difficult to Cure?), Van Halen's eruption bit, Ozzy's (well, Randy's) Mr Crowley intro, even that Am I Evil? lick at the beginning, just from the top of my head. But all these are just small things, little tastes of classical flavour. And still way more prominent than any Metallica bit (again, save To Live is to Die). And of course this is not even touching upon the prog rock and symphonic rock stuff of the 70s, which did feature actual classically trained pianists as keyboard players for example. But Malmsteen, man, he just totally fused the two, essentially created the hybrid, as you put it.

I love Cliff Burton more than most people and I have a VERY, VERY soft spot for him and I always say that without him Metallica would never have accomplished 1/10th of what they did. But to claim he (and the band) brought classical music to metal is waaaaay off the mark IMO.