In regards to Master of Puppets' production, it's always worth remembering the context of the time. Back in 1986, having a thrash metal album even as polished as that was basically unheard of at the time. That, alongside probably Reign in Blood's sound, heralded a kind of coming of age moment for the genre. It showed that aggressive music didn't have to wallow in that kind of straightforward underground fuzziness that Venom or Celtic Frost embodied. Maybe it sounds antiquated now, but Fleming Rasmussen's work on that album was pretty revolutionary for the time because while Ride the Lightning certainly has a distinctive energy to its sound, it doesn't match the cleanliness and tightness of its follow-up.
As Kev pointed out too, even though it doesn't sound as polished as The Black Album, it still has its own raw nostalgic charm that's admittedly hard to pinpoint, but newer metal bands that adopt that kind of earthy raw crunch while retaining clarity (such as Horrendous or Power Trip) show that the appeal isn't just from rose tinted glasses. 80s Metallica in general was a very good illustration of how just because metal got heavier, it doesn't mean it couldn't still retain its sophistication. I think Death pretty much did the equivalent for death metal in the 90s too.
One style of production I do not get however is Pantera's (which is also probably my pick here), pushing what were perhaps the signature elements of Metallica's production values at the time (scooped guitar, sonically tight drum sound, distorted bass etc.) and pushing it to earbleeding extremes to the point where the trends that spawned from them basically ruined mainstream metal production (at least within that kind of groove/thrash metal side of things) for a solid decade or so. The whole "aight, let's remove the all mids because we want crunch" approach is basically laughed at nowadays in producer circles from what I can tell and with good reason, it's needlessly abrasive and has aged incredibly poorly.
Other than that, I kinda understand Pantera's influence musically in terms of taking the template of thrash and adding more mid paced grooves and syncopation and bringing that to the big leagues (given that Exhorder had sorta done something similar already), although I will struggle to support any band that openly brands the Confederate / Virginia battle flag and provides a platform for some uhh... questionable talking points with regards to some of Phil's rants, as much as I'd want to separate politics and personalities from the music.