the Labour party in the UK is absolutely a leftist party. If they ran as anti trans, and I absolutely don't believe them when they say that because they were the primary supporter of that trans group (mermaids I think it was called), they would've done so because they know it isn't actually supported by the wider population of the UK and they need votes. The Labour Party used to be run by Jeremy Corbyn who was a radical communist that eventually got ejected by the party because of open support of terrorist organizations and anti-semitism. Kier Starmer, the current leader of the Labour Party is a socialist, human rights lawyer who's early career included going to Jamaica to sincerely defend rapists and murderers (and not in a "well its my job as a defense lawyer" kind of way). The only party in the UK that's further left than the labour party are the Greens and whatever party Corbyn and that one crazy lady just started in hilarious fashion where people couldn't even figure out the name of the party (because the website was "myparty" or something like that, and the actual party isn't called that).
Saying that the Labour Party isn't leftwing because other leftists hate them isn't an argument because the left eats itself over purity tests all the time. It's practically the main facet of being a leftist and something the right makes fun of all the time. Walking on eggshells over every issue, just to find out that you're not quite left enough. If I'm not wrong, Jordan Peterson said exactly that to Ethan Klein at one point, and a few years later, he was proven right about that.
You can tie this as well to that essay you linked, because as far as I'm aware, No Mercy was the game that got removed on every storefront over essentially being a game about revenge sexual assault. And while I don't agree with censoring of it, that had nothing to do with the LGBT, and that was also a feminist organization. Regardless of what that groups wider opinion on the LGBT is, it's still another subsect of the left, against other subsects of the left (and others). And a fun fact, because everyone seems to immediately think "Christian = conservatism", there are a lot of incredibly left wing Christians. See the entirety of the Anglican (UK)/Episcopalian (USA) churches, and also the Methodist Church, which eventually had a schism into the Methodist and United Methodist churches over that very distinction.