What I don't get is how you're meant to initiate a conversation with a woman when they know, and you know, and everyone knows that in doing so you are signalling your desire to have sex with her.

Like that's the only reason anyone could ever have for going out of their way to talk to a member of the opposite sex who is in their age range. You wouldn't complement someone's jewellery or ask them about the band on their t-shirt or say ANYTHING to them unless you wanted to break the ice, to open a diaglogue, to lay the foundations of a possible eventual sexual encounter. No one goes up to ugly, unnattractive people and strikes up a conversation with them about some inane hobby they have, because the attention of an ugly person means nothing to anyone.

So with all that said, how does anyone initiate encounters with women without it being openly transparent that the only reason for doing so is sexual attraction? How is that not cringey and weird?

there were so many fit birds in town lads I can't even tell you

it's just mental how many attractive women exist. It's cruel really. It would be better that I were dead so that I didn't know they are all out there.