I like your optimism, but given Labour under Starmer have expelled or repelled many on the left, I suspect that Starmer's not thinking 'Now I have a comfortable majority I can shift the Overton window and politics in general leftwards'.
He, and Reeves, made a big point of campaigning to convince big business that the economy would be safe in their hands, and that they would effectively maintain the course the Conservatives were steering.
Personally, I'd be delighted if Starmer turned round and said 'Now that I'm in office, and bearing in mind the short and long term impacts of children living in poverty, on their education and health and life chances, I've decided to remove the two child cap on child benefits.'
Also because children shouldn't be punished for their parents actions, ffs.
But I don't think he cares was leftists think. And I don't think he's going to be influenced by the Greens. He could comfortably face the prospect of losing a dozen or more seats to the Greens at the next election and not lose any sleep over it.