Could I have won this position?

no i dont think so a draw seems like a fine result dont really have any tactics for either side.

 

i can't use Nalimov Tablebases due the number of pices (8 in total) but after lets say blacks h pawn traded for one of the whites g pawns, its draw. but let me tell you that rock endings are one of the most tricky ones and its really hard to play them right cose some times the good one its a one really counter-intuitive, so if you have confidence on your engame knowledge,  its good to go for it till the end ive won countless game's that were actually drawn ones but to accept the draw your opponent first have to prove it to you.

It's pretty dead.  Even if you win the pawn on g4 it's still drawn.

draw

One of the rules to remember is that often in R + P endings if the pawns are on the same side it is often a draw even if it is 3 vs 2 pawns.