I think it's more "When was the last person this over so organically?" rather than being for a catchphrase or moment.
Austin 3:16 was a moment, it worked and Austin exploded but had Pillman not had his crash he'd have done something similar (or probably more extreme) instead... it would have worked equally well because the fans were ready for that. Same for Punk's pipebomb or even Hogan's win over Sheiky Baby...
Shawn was forced into the position over time, as were guys like Kurt, Eddie, Rey, even Cena...
The guys I can think of who organically got over in spite of the E's wishes are:-
Jeff Hardy - This was happening as early as his win over Trips a few years ago, the fans wanted him but Vince didn't want to give him it... Over time he continued to build that momentum until Vince HAD to give it to him.
Bret Hart - Bret was able to parlay his skills into getting over. He was never intended to be "the guy" but during the 80's Bret was often the most popular guy other than your Hogan/Warrior/Savage. He had a big assist from Gorilla Monsoon with the "Excellence Of Execution" line but when he finally got the big belt in those bizarre circumstances it didn't feel like "Flair refusing to job on TV and only to a good worker" more that Bret had done the impossible and he never looked back. He was revered genuinely all over the world for a time...sadly once he started believing his own hype it was the beginning of the end but for those early - late 90's he was "the people's champ."
Christian - For a time he was getting insane reactions, and it's easy to put it down to Edge's retirement. But it goes further back, Christian was always able to get a strong crowd reaction but Vince HATED it. Why we will probably never know, but when he finally won that title it wasn't cos he'd been pushed to it, it was similar to Hardy because the fans had "made it happen".
The winner though is:-
Mick Foley - There is never a guy who has been built as organically as Mick was nor will be again. Forget the Cell, that was just the icing on the cake. He'd already taken a potentially career killing character (Manson the Mutilator?) and made the fans care enough about him that they were chanting for him during that Cell match and many wanted him to win. Even once the bumps were out of the way and the dust cleared people still were buying into Mick Foley and not cos he got hurt a lot... it was cos he was clearly a special talent. That he could be in the Rumble as 3 different characters helped him, that he could make something as goofy as jabbing his legs with a pencil or a sock a high spot... or that people just "fell in love" with Mick Foley the guy. Not literally but as their "favourite", not just the underdog but the closest they have to an avatar for themselves. Why was Tony S's "Butt's on seats" line so crucial, because to so many watch Foley was their favourite WWE guy... They were watching Nitro cos they didn't buy Rock or Austin but they all GOT Foley and seeing him win the title was their moment where they could say "We did it..."
This is what Bryan taps into so well, he looks like one of us, is goofy, a bit of a "dork" even but regardless of chanting Yes or No he is able to make the fans love him in a way few have done for just being himself out there...
Vince knows it, Trips knows it... I am sure they'd rather it was someone else... but it isn't and we're gonna start getting the payoff to the Bryan story soon. If anyone is gonna encourage more mainstream attention it's Bryan winning the title and being the face... cos he is the "anti-sports entertainer" who does it better than anyone else... If anyone can convince casual or non fans to give it another try it's the most "non typical wrestler"...