The trouble is it’s not really “the right” or “the left” that want that - it’s the extremes. But neither the Tories nor Labour have been able to cogently oppose the extremists. With the Tories it’s the natural consequence of UKIP terrifying the shit out of them. They got a tiny percentage of the vote, but it was 80-90% directly pilfering the Tory vote. With Labour it’s unfortunate timing, and the legacy of Blair and Brown, in suppressing the left of the part, that lead to it suddenly becoming a huge thing (I imagine Zephro can say more and better on this).
The entire referendum was a crude Tory attempt to use a wedge issue to simultaneously hurt Labour and reduce the threat of UKIP, as per the instructions/methodology of Lynton Crosby. He’d used similar tactics before and EU membership, particularly immigration, was indeed a wedge issue in Labour, because a sizable number of Labour voters are skeptical about letting people in to “take their jobs” (even if party members rarely share this opinion).
I blame Gordon Brown for the whole thing though. Hear me out!
It was he, who, with his “bigoted woman” crack, and then his failure to take a stand (instead becoming servile and apologetic to a woman who was, undoubtedly and clearly, a bigoted scoundrel), made it clear that this was a serious wedge issue. Before that the Tories were paying no attention to that - they seemed to think only they had this as an issue.
So by being, in sequence, an idiot, and then a coward, he started the ball rolling. Had Labour taken a firmly “No fuck this shit” stance back then, it’s unlikely Crosby would have tried that angle.
Labour seems to be keen to create a new wedge issue for itself right now with the whole facepalm-y “BAN PRIVATE SCHOOLS!!!” they’re pushing extremely hard whilst simultaneously attempting to soft-pedal (great combo…). There’s no possibility this ends well for them. They missed their chance in 1965.
Of course Labour has a long history of self-sabotage, in and out of government, that the other parties just don’t share to the same degree (albeit the LibDems were the most spectacular in their self-sabotage, but that was more of a one-off Nick Clegg thing), so I suppose I shouldn’t be surprised.
And I know there a plenty of cynical Labour types on both sides of the party who are fine with Corbyn losing, because they’re convinced Boris + the lunatics will be a total disaster as they try to mis-manage post-no-deal Britain (which is fair, they will) and expect 2+ terms of government after that (which is, I think, reasonable if they get a good candidate).