I think Critical Role's success and D&D 5E's success went hand-in-hand. Each one helped the other.
Switching to 5E helped Critical Role because CritRole was a completely new show that needed something to hook viewers. Using the current "big thing" in roleplaying... D&D 5E... was that thing. D&D as a brand was a more known property to all those potential Geek & Sundry viewers that perhaps were board and video gamers but not necessarily into RPGs. Had CritRole used Pathfinder, there is no guarantee that it would have caught as many casual Geek & Sundry viewers because less of them would have known what "Pathfinder" was. They'd have had to have been told "Well, it's like Dungeons & Dragons, but it's a little different..." yadda yadda yadda. But by actually using D&D there was less instruction needed. This probably helped onboard more people into the show and into this particular game and roleplaying in general.
5E of course benefitted because CritRole was the right group with the right players and the right DM at the right time to show people what was possible to be done with this game and who were willing to take the game seriously. D&D and the art of roleplaying were not treated like it was a joke that was being played. I don't think the same can be said as much with regards to say Adventure Zone or Nerd Poker or the Penny Arcade / PvP podcasts that were all active by the time CR started. All of them were shows with silliness to them and where the gag was almost "Hey, look at this goofy thing we're doing!" rather than treating the player of the game seriously but letting the humor come out of the play. I mean heck... I think all three of those podcasts had characters with joke names, and that right there told us that they were not treating their games like they were anything more than a fun, goofy time.
CR didn't treat things that way nearly as much-- at least not on stream. Mercer treated the stories as serious and the players treated Mercer's stories as serious. And I think this is because they were all trained actors in improvisation... where one of the central tenets is to treat scenes as real, even if all kinds of ridiculousness is going on around you. You wink at the audience in the middle of a scene to tell them "Yeah, I know this is stupid!", and the entirety of the scene breaks down and you lose them. CR didn't do that. They treated their stories and the game seriously while also doing all kinds of goofy stuff in character. And thus it made it easier to hold an audience week after week because the cast was treating it real, and they never made the audience feel the fool for watching them do it.
Thus... with all these players being able to be seen live and onscreen in-person taking this game seriously and not treat it as a big joke, it allowed for more people to not see D&D as a joke... which quite frankly has been its most difficult stumbling block for decades. Seeing it played allowed people to understand the game was actually as serious as you wanted to make it. And that's what helped 5E and all the RPGs out there get a foot into a door that had up to that point been held shut by a large swathe of culture. D&D was no longer this gag of a game because you could now see it played just as seriously as any other game out there.
