To repeat the obvious, the 'centre' shifts over time. For example, much as I disagree, currently I would say the 'centre' is still pro-Brexit ('Get Brexit Done' was very appealing, even to some Remainers). But it was mildly pro-EU before and may shift again.
Similarly I think a lot of 'sensible' opinion is now that austerity was taken too far, even if at the time the 'centre' view was that it was sensible. You don't hear Rishi Sunak arguing that that 'There's No Magic Money Tree'. And further in the past the centre would have also been fairly openly racist, sexist and homophobic. It took people willing to go beyond the existing consensus to shift.
This is simple stuff and it is not intended as a justification of my own politics. It is evident that someone like Nigel Farage did not think the way to achieve his political objectives was to tailor them to where public opinion already was.
A year or two back I found a good take on the process of political change, or the lack of it, in this book. Yudkowsksy is not a leftie, I think he'd be a pretty moderate US liberal. But the explanation is a good one.
"The... force locking bad political systems into place is an equilibrium of silence about policies that aren't 'serious'.
"[I]f an existing politician talks about a policy outside what journalists think is appealing to voters, the journalists think the politician has committed a gaffe, and they write about this sports blunder by the politician, and the actual voters take their cues from that. So no politician talks about things that a journalist thinks that a journalist believes it would be a blunder to talk about. The space of what it isn't a 'blunder' for a politician to talk about is conventionally termed the 'Overton Window'...
"To name a recent example from the United States, it explains how, one year, gay marriage is a taboo topic, and then all of a sudden there's a huge upswing in everyone being allowed to talk about it for the first time and shortly afterwards it's a done deal. If you suppose that a huge number of people really did hate gay marriage deep down, or that all the politicians mouthing off about the sanctity of marriage were engaged in a dark conspiracy, then why the sudden change?
"With my more complicated model, we can say, 'An increasing number of people over time thought that gay marriage was pretty much okay. But while that group didn't have a majority, journalists modelled a gay marriage endorsement as a 'gaffe' or 'unelectable', something they'd write about in the sports-coverage overtone as a blunder by the other team.
"...Those journalists weren't consciously deciding the equilibrium. The journalists were writing 'serious' articles, i.e., articles abut Alice and Bob rather than Carol. The equilibrium consisted of the journalists writing sports coverage of elections, where everything is viewed through the lens of a zero-sum competition for votes between Alice's team and Bob's team. Viewed through that lens, the journalists thought a gay marriage endorsement would be a blunder. And if you do something that enough journalists think is a political blunder, it is a political blunder. The journalists' sports coverage will describe you as an incompetent politician, and primates instinctively want to ally with likely winners. Which meant the equilibrium could have a sharp tip over point, without most of the actual population changing their minds sharply about gay marriage in that particular year. The support level went over a threshold where somebody tested the waters and got away with it, and journalists began to suspect it wasn't a political blunder to support gay marriage, which let more politicians speak and get away with it, and then the change in belief about what was inside the Overton window snowballed."
I think this is pretty much spot on. (It also sorta mirrors the Keynes line about stock-picking becoming about second-guessing other people's second guesses about other people's second guesses).
We see the process happening all the time. Anti-EU feeling was on the fringe for a long time, but obviously ticked up significantly. Yet even when (in retrospect) it was tipping over into a majority position it was treated as 'loony', even by many Conservatives. And it is still treated that way by many. Similarly, you can see that the media-political class isn't completely sure about where to position itself in relation to Black Lives Matter, or at least the more radical end of its activity like toppling statues.
As an aside, I think a defensive political lesson to take from this is to listen to the 'cranks' both on your own side and the other's, and to listen to the best versions of the arguments they are making, not focus on the obviously idiotic ones. Today's 'crank' could be shaping tomorrow's centre, but today's centre will dismiss her as a 'loony', 'Trot' etc.
To go back to the beginning, what I think I find frustrating about 'you can only win from the centre' is that it is takes the equilibrium for granted. It's playing the same kind of role of as the journalist in Yudkowsky's scenario - policing the boundaries as they are currently understood. And essentially it betrays a conservative approach (I genuinely don't mean 'conservative' in a pejorative sense, it's an entirely legitimate way to view the world, I just don't share it). I genuinely believe there is more freedom of manoeuvre than it allows. Certainly I think there is scope for economic radicalism that is denied by 'win from the centre' advocates.
I'll finish up with a quote I heard from a union mate that apparently comes from the well-known US union organiser Tom Woodruff: "A rolling programme builds its own consensus. Consensus by itself never built anything but worthless shit."