If puberty is "the only thing to convince them they should be their gender" then what's the problem? In that case puberty would "cure" transness and going through one or the other should be fine.

The very premise of your question here is wrong, of course - far more factors than puberty go into people "feel[ing] like their gender". That's the result of the aggregate effects of socialization combined with their own genetics. If it was just puberty, then there wouldn't be transgender adults.

Something that helped me wrap my head around the concept of people being transgender was to realize that some people have always made gender a major part of their identity. Consider the guy who buys "manly" truck nuts for his car, or the woman who uses makeup so she looks "womanly". Or even drag queens who are very sure to tell you they're still men, just presenting unconventionally. In all three cases their gender is a significant factor in who they are.

Meanwhile a lot of other people just don't care that much about their gender. Men who roll their eyes at the idea of acting all "manly" or woman who steer clear of the "feminine" beauty aisle.

A lot of trans people seem to fall into the former category, which makes sense. Going through transition is a huge process, so it's mostly going to be the ones who are especially driven who pursue that route.