Yes - I have definitely become a better climber as I lose weight. That said I still suck at climbing, but now I put the 190 version or 200 or 210 version of my self to shame on the hills now.
I am currently 179 on my way to 165 at some point...at 165 if I still suck at hills I'll just blame the bike or something else.

BTW - just about everyone has to get better at climbing to get to the "next level". You're not alone.
Also...lose the weight the right way or you'll suck more at climbing. If you sacrifice muscle during weight loss....