Only way to make support more appealing to the greater playerbase is to make healing less important, and push all supports into the DPS/Healer hybrid role.
This issue isn’t that Blizzard have done something wrong, the issue is that most FPS players don’t enjoy healing or buffing their teammates. And this extends past FPS to most action games. At least in my experience, in literally every single game that has different roles, DPS is the most popular and has the longest queues and Support/tank roles having much shorter queues because they’re in much lower supply with any pure tank roles being absolute dead last in popularity.
Even games without a strict role queue, like RB6, everyone loads up the game and just instalocks the Fragger characters and either refuse to play a supportive operator or begrudgingly accept that they were last to the draw. The only time this is flipped is when a supportive operator is so essential to the team because of an overbearing ability.
SMITE: Before role queue, it was a race to Jungle, ADC and Mid lane roles (the high damage ones) with most people preferring to pass up the tank and the support role.
Elder Scrolls Online: despite there being 2 Damage dealer slots per team and 1 tank and 1 healer, tank queues are instant, healer queues are short and DD queues can easily be 40+ mins.
Evolve: very short queues for Medic and Trapper roles, since they are low damage roles. Longer queues for Support (which in this game were sort of comparable to Overwatchs off-tanks) and assault, the pure damage dealer was the second longest queue. The absolute longest queues by a mile where the Monster queues. The teamless killing machine role.
I could continue for days. Where role queues aren’t enforced, people will always race to the damage roles. When it is, damage dealer queues skyrocket. Its not a Blizzard issue, its a player issue that you cannot eradicate without homogenising the game.