Moltres, Dragonite, Tornadus, Zapdos, Thund-T, and Volcarona, are the only Pokemon that would get a notable buff in a metagame which SR doesn't exist (I believe the stats of the SR-less ladder show this pretty well), and they're pretty easy to handle, regardless of whether or not SR exists. The problems with the aforementioned Pokemon lie in their stats; Moltres, Dragonite, and Zapdos are rather slow, once you take into consideration what their intended tasks are; Tornadus is very frail; and Volcarona isn't going to find time to set up and comfortably sweep vs the average team (i.e, the player has to work for boosting AND cleaning with Urugamosu). Also, I think that people tend to underestimate spinning. It's pretty easy to spin in BW2 OU, thanks to rain, really (Starmie, Tentacruel), and the accessibility of every Foresight spinner in the game. That said, SR keeps a nice tight lid on stall, ironically, as it gives offensive teams an easy means of pressuring key defensive threats, such as Chansey and Jellicent, and punishing mindless switching without the need of risking the Pokemon currently in play. Defensive teams in general become much more skill-reliant and harder to successfully create with the existence of SR, which is really freaking good considering that new and veteran players alike prefer fast-paced games.
Also, Fire-type Pokemon were horrible long before SR was introduced. RBY prefers paralysis and freeze status due to critical hits being reliant on Speed and freeze status being permanent in that gen. Though Fire-type moves didn't thaw in that gen, burning a not-Golem/Rhydon Pokemon meant that the same Pokemon wouldn't be debilitated by doubtlessly more harmful status later on in the game. Sure, Fire-type Pokemon had good match-ups with key threats like... ok, I can't think of any, aside from Exeggutor and Jynx (both of which will be sleeping stuff, anyway), that are even in the slightest bit relevant. Moreover, Fire-type Pokemon in RBY are generally the same utility-wise; SD Charizard is an OK lead, but it shares the same premise as, say, Magmar, or Arcanine (warding off Jynx / Eggy leads while also breaking Chansey). Finally, and most obvious, is the simple fact that Fire-type Pokemon would attract huge threats like Amnesia Slowbro / Snorlax and Golem / Rhydon--and these are not Pokemon one should give any sort of momentum to.
GSC wasn't too kind to Fire-type Pokemon either, as the introduction of many more special sponges *cough*Snorlax*cough* made Fire-types more of a liability to have on a team. I think that the only remotely good one in GSC was Charizard (the only BDer that can immediately threaten Skarm), but even that turned up pretty bad once people realized how simple Charizard is. RSE continued this trend with the introduction, buffing, and eventual popularization of bulky waters. Blaziken is literally the only Fire-type that didn't struggle in RSE, as it, thanks to its marvelous dual STAB, could smash common defensive cores--and it has two extra moveslots to do whatever with, making it versatile enough for players to think twice before trying to combat it. IF ANYTHING, DPP buffed Fire typing thanks to the physical / special split, and the debut of Heatran and Infernape; it didn't make Fire-types of yestergen any worse than they already were. Then BW[2] gave us things like DroughtTales, Victini, Nattorei, and Hurricane Moltres, making Fire-types a tad more useful and threatening. I suppose that fan favorites like Charizard wouldn't suck quite as hard (it would still be pretty ****ty thanks to, you know, rain, sand, and its slow speed + defenses) without SR, but, again, keeping rocks off of the field isn't as difficult as people tend to make it seem.
tl;dr, Fire-type Pokemon suffered due to bad design, not SR.
