Honestly, those systems don't look too good for trans people. The NHS requires you to wait years to go on $40/month hormone replacement therapy. The government passed a law banning prescription of puberty blockers for kids that are undecided on which gender they'd like to be as adults, forcing them to go through the wrong puberty or work around the system. Saw a post on Reddit yesterday that was "I support my daughter but we can't keep her on GnRH analogues anymore because it's illegal here." The consensus was to fly to Spain once a month and pay cash for the injection, or skip puberty blockers and start estrogen early.
This is not amazing. The single-payer model is unfortunately too political, and somehow being trans is a hot button political issue right now. It really, really sucks for these kids.
Meanwhile in the US, there is a lot of hate, but a lot of insurance companies do cover the necessary care. They even pay for flights/hotels in other states when care isn't available in your state because of political issues. We all like to hate insurance, but seeing those things commonly available in private insurance policies is ... refreshing. You are still going to have to file appeals, but it's better than "the government doesn't believe in trans people so fuck you".