Your short game skills might be more or less important to your scoring, depending on your skill level.
It depends who you are
Short game importance scales with how often you use it. A Tour player misses a handful of greens a round. A twenty handicap misses most of them.
So the same advice cannot serve both. For the higher handicapper, chipping and bunker play carry real weight simply through volume.
The catch
Even for that player, short game is rarely the biggest single bucket. Full swing shots outnumber short game shots by a wide margin, and the worst full swing outcomes cost far more per shot.
Short game is where you stop bogeys becoming doubles. Long game is where pars become birdies.
How to decide for yourself
Track up and down percentage and where your missed greens leave you. If you are converting less than one in four from around the green, the short game work is justified.
If you convert well but still score poorly, the problem is upstream and no amount of chipping practice will fix it.