It's a long video, but this video by Philosophy Tube explains why it's definitely a trans rights problem, not just the broken NHS. The long wait times across the board is what makes the trans healthcare side of it so outrageous, because it's structured in a way that takes an already bad problem (shortage of services) and makes it so much worse through gatekeeping and bureaucracy.

Your comment also inadvertently highlights one of the especially insidious aspects of UK trans healthcare being broken and nonsense — it's fairly common knowledge that the NHS is on its knees, so activism aimed at making trans healthcare less shit is often perceived as an attempt to "jump the queue" so to speak, which feeds into anti-trans sentiment. (To be clear, no negativity is intended towards you here, I think that unless you have direct experience of trans healthcare in the UK, it's hard to know just how bad it is).

I was going to give a couple of examples of the kind of gatekeeping I mean, but I'm bad at brevity; if you're someone who watches video essays, I would strongly recommend the Philosophy Tube one above. It's even pretty entertaining, for such a grim topic.