Perhaps we are obsessed with learning words because it's the easier problem to solve.
I think that depends on the person and/or language.
My Polish grammar knowledge is now pretty impressive, but I'm having a terrible time trying to remember words, so spend the majority of my time attempting to force memorisation with flash cards.
In a way it's kinda fun - I can go through pages of grammar exercises, conjugating and declining correctly, but with no idea of what Marta did in the past with 101 of somebody's somethings.