If you take a longer view, I think you'll conclude differently, go back to Atlee and you'll see the lurch back and forth between Labour and the Tories follows the same pattern of returning to the centre ground. The last time Labour moved this far left, we got Thatcher. Thatcher pushed everything to the right, Major attempted a centre ground reform and lost out to Blair. Brown pushed everything left, and Cameron came in to centre-ground. The Tories went right after the Hague reforms and Labour went left after Milliband changed procedure. We've now had an absolute cavalcade of thundercunts pushing everything rightwards in the Tories, followed by Sunak who is attempting to portray himself as a respectable centre-ground politician and simultaneously appeal to the hard right of his party. He's failing, the way Major did and we'll have a centre ground Labour Party. Starmer has moved the Labour Party back into electability.
You want to know where the big reforms for LGBTQ+ people come from? The Labour government decriminalising homosexuality in the 1960s, the Blair Government enshrining the Human Rights Act (1997), reforms on NHS treatment between 1997 and 2010 and (oddly enough) the centre-right Conservatives legalising LGBTQ+ marriage. Progress is made in the centre ground and undone at the extremes.
And to your point, I don't think Starmer is using Trans people, I think he was appalled by Sunak's comments (like most people were) and I think he said so. I think he's broadly supportive of change, but understands this is a fight best had at a different time and when in Government. Labour has a habit of making itself unelectable by having a civil war whenever it looks like it's making progress towards power.
If you zoom out and take the long view, there has never been a safer time to be Trans in the UK (I sincerely believe that) progress is slow but steady and I am confident a centre left coalition will make significant progress. It might not feel like it, but politics moves in years and social media moves in minutes.