Maybe - to your point, if we think of happiness as like “living one’s own purpose fully”, then yes, it does very directly.

But I was referring to happiness more generally as enjoyment, joy, satisfaction - that type of thing.

And in that case - there are plenty of ways being better = less happy. Eg if I were to sacrifice myself to save my family, then that’s the best version of me, but I’d be pretty dang unhappy about it.