I have recently come around to viewing the game in a bit of a different way.
Let's say you're playing DPS. You're going to put out a certain amount of damage based on the character that you're playing, and most people within the thicker parts of the skill bell curve are going to do roughly similar amounts of damage over time; there isn't fundamentally too much you can do to improve your total damage dealt for a given game if you're already doing things like showing up to the fight, ulting at vaguely appropriate times, and achieving good accuracy numbers. What you can do to differentiate yourself from what other players would achieve in the same scenario and playing the same character boils down to two things:
1. Focus your damage in the right place and
2. Take as few unnecessary deaths as possible
As it turns out, these are rather difficult things to do but it's helped me stay focused on the right things to do instead of, say, firing pointless damage into a full health Zarya.
If we're talking relevant stats, I think Deaths and Win Rate are the ones. Assists are really important for Supports, too.
The way I improved at support was to stat-whore assists, not being a passive support.
Assists are a decent stat and deaths have some relevance to support players but I'd say they're pretty useless for just about everyone else. Ideal tanking involves deliberately putting yourself in harm's way and giving it up to the healers to keep you alive; if you're playing optimally in a situation but the healers aren't, it's you who's going to take the death. Similarly, flankers who make a lot of risky plays are going to die more than more passive ones but are probably going to get more wins (to an extent) due to more opportunities for playmaking. Yes, you could just alternatively become an unkillable god and successfully flank every time but that's not really realistic.
If I were a Mercy player, though, I'd care a lot about deaths. Her character is based more or less entirely around uptime since while you're alive your output is more or less fixed and while you're dead it's 0.
Like the other stats, even win rate can vary a lot from season to season without the way you play changing. They are just different games with different teams.
This is only true to a certain degree. The more games you play, the more likely your win rate is to reflect your ability to put a game in the W column assuming you don't experience any significant and abrupt changes in skill level (which is possible although this is probably something you'd notice). Any player with a significant number of games played - let's say 200, but really I'm just making that number up - will have, in all likelihood, experienced a roughly equal number of throwers/leavers on and off their team and will have played enough games that their win rate over time is fairly stable.
...That's the idea, anyway. The SR system really fucks this up by "guessing" players into places they shouldn't be. I'm at around 200 games now and my win rate is still far from stable. This creates weird situations where it's hard to use win rate because, while I'm at 3900 or so and my win rate is 60%, my win rate isn't 60%
at 3900; that number includes a lot of games played well below where my rank should be because of my low placement. For players who drop, the opposite happens: A player with a career high much above yours with a poor win rate may nonetheless be better at winning than you at your common rating.
Regardless, it's still by far the most useful stat by itself, even if it's not especially useful.
