The point I'd make here is that "centrist" Labour parties who tend towards the right on a bunch of key policies aren't going to do this. Barring 1997-2001 when the balance of power between the soft-left and the Blairites in the PLP was such that the former had a lot of influence over policy, New Labour's time in government didn't see the centre ground move to the left. Quite the opposite, they ceded a lot of key ideological ground (on things like privatisation and deregulation of the financial sector) to the right from the start, and moved rightwards from there (particularly in terms of social policy like crime, welfare, immigration and asylum) as Blair's more reactionary faction started to dominate the PLP after 2001.
Starmer is starting off way to the right of where Labour were in 1997, and nothing he's said or done suggests he has any intention to move leftwards in government.