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.

I agree, I'm not too big on the John Bush-era either. I did love 'We've Come for You All' though. I know everyone has their own preferences, I've just never understood why Anthrax gets the least notice of the big four   

I don't understand why people think Anthrax is the worst of the big four often, they are very underrated. Anthrax is my personal favorite of the big four.  

They were very important in the development in thrash and speed metal, just listen to Spreading the Disease.

As for Mustaine's influence, he sure had a hand in shaping thrash metal from the start but the heaviest songs (like Whiplash, the final part of No Remorse, Motorbreath, the Hit the Lights riff) weren't his with the exception of Metal Militia. He wrote Jump in the Fire, Four Horsemen (as Mechanix) and Phantom Lord which are the closest to "traditional" heavy metal. His statements that he created thrash are really self-flattering as he always mentions the solos as being an integral part of it. Hetfield was way more responsible and it can be confirmed with the host of GBH and Discharge shirts he was seen wearing afterwards. He did not know those bands before Kill Em All was released but it is telling that he really appreciated the other extreme stuff of the time. 

Also let us not forget that Dave Mustaine was calling Megadeth the state of the art speed metal band, he wanted to be the fastest and the most technical (and certainly faster and more technical than Metallica, to prove a point).

Anthrax is VERY influential. Perhaps some don't like the kind of influence they had but it is undeniable. For starters, SOD was a HUGE influence to the groove/nu/rap/modern/whatever metal. How many bands have ripped off March of the SOD? That's 1985! The crossover, the image, the skate culture, the I'm the Man single which catapulted them to success, their tour with Public Enemy (unheard of! hip hop and thrash metal?). They really sold more than Slayer. As did Megadeth. 

There is a nice story about all that during the Clash of the Titans tour, when Mustaine was wondering why Slayer got to co-headline (and thus play last some nights) since they were clearly less popular than both Megadeth and Anthrax. And Scott Ian kinda agreed with that. Then famously, Slayer blew both bands out of the water on stage and in the merch sales, which resulted to the other two asking them to play last every night because otherwise the crowd was too exhausted for their shows. And again, famously, Kerry King was saying "serves them fuckers right, they wanted to rotate, we'll rotate!"

Good times. The least popular thrash metal band out of the big 4 destroying live the other two. Metallica of course in 1991 were on a completely different level.