to be honest, symmetra should probably be listed as a defense character rather than a support
That's a bit like saying Torb should be a Support because he can make Armor packs. It's a good ability, but it's not what you get a Torb for.
Heroes are classified based on what role they take in a fight, and their ultimate. Defense heroes take an active role in the fight, and are good at staying in a fight for a long time. Symmetra CAN fight, but she's not expected to fight, and while she HAS turrets, they're all but expected to die quickly. Symmetra's defining characteristic is her Teleporter, which is clearly a Support ultimate. It's powerful enough that her other abilities are made relatively weak because they serve primarily as a means to charge it. This is why Symmetra is commonly used on the first cap but not the second in competitive play -- her turrets aren't reliable, so she can't be used as a main Defense hero, but her teleporter can be extremely useful, so she's used on levels where the Defense needs to defend a point that is far away from where they spawn.
Defense isn't "good at staying in a fight for a long time". That's a Tank. Defense is good at area denial and holding a location. By the official definition of the Support, "Support heroes can heal allies, enhance their team's abilities, and debuff enemies.", and Defense, "Defense heroes are good at guarding locations, creating choke points, and preventing objectives from being taken.", roles, Symettra is far more a defense hero.
D.VA had some issues as a Tank, as well, until her recent buff.