And a lot of old-school lefties in the UK consider Blair's New Labour to be centre-right, "Red Tories" etc. Blair might have been left-wing but he certainly wasn't a socialist.
The "left" in Britain arguably haven't won an election since 1974.
It has to be considered that "the left" has been structurally divided in three parties (Lab, LibDem, Greens) for most of this period, which is a cardinal sin in the British system.
The right had fewer splits, and arguably none of them is comparably structural.
The left might have lost a lot of elections, but I think you can still make a case for the original point the "right-of-centre" are a minority in the UK, even if it's only a narrow majority. The Conservatives haven't won a majority of the popular vote in nearly 90 years (but then neither has any party)
The "left" in Britain arguably haven't won an election since 1974.