I feel part of the problem stems from her ability to fire her rockets, and at the same time spamming her primary fire (which has no penalty other than slowing down her movement). Like you said, she can do too much.
She can literally chase the various high-mobility heroes across the map with her 3 sec. boosters on a short cooldown, burst down any 200 HP hero with next to no ability for counterplay, and completely delete incoming damage.
1. The main issue with DVA is defense matrix.
I personally have no issues with her Defense Matrix as it is, and what it used to be before she got her rockets, and removing it, at least to me, feels unnecessary.
2. DVA provides a much necessary role against flankers and, more importantly, Pharrah
This is more an issue caused by the stereotypical sh*tty DPS players and can’t reasonably be attributed to d.Va. Almost every game that has a Pharah in it sees Pharah ignored by the DPS players, whose job, among many others, is to keep the skies clear with their hitscan weaponry. This forces d.Va to be used as anti-air, which isn’t her job.
3. More importantly: as the topic says, DVA is a tank but yet consistently punished for being one
To me, d.Va feels like a tank with too much mobility, which, by extension, forces her into a role that doesn’t really fit her.
4. Her high pick rate is because she’s the only true mobile tank capable of challenging heroes with range
Not sure where I stand on this. I get your point, but I can’t help but feel like your team should counterswap if they have issues with ranged heroes, instead of leaving it to the d.Va to try and protect them from the ranged heroes and at the same time, peel for the team.
And this makes it difficult to come up with a fix, as there’s both a few issues with her current kit, but also with the mentality of the player base (and this, unfortunately, won’t change anytime soon).