I could never really get into TF2, but I love Overwatch. One of the brilliant things about is that there are many classes with a wide variety of play styles including several that don't require great aim or super human reflexes. One of my favourite heroes is the support Mercy where you don't need to aim much at all, but you do need to have a good awareness of what's going on all around you.
Characters range from needing to be pinpoint or have great tracking like with Widowmaker and Genji respectively, to characters that almost don't need to aim, like Winston (who's primary fire arcs in a radius), Reaper (who has a very wide spread), Reinhardt (who's main purpose is to shield while his attacks are all very generous in forward arcs), and the supports like Mercy, Symmetra, and to a lesser extent, Lucio. Maybe a quarter of the cast doesn't require much at all for aim and many are quite forgiving in terms of reflexes. Many of those classes also form the core of your team.
It's really crazy how positive the game feels: even losing several matches in a row doesn't feel bad with how they carefully craft their feedback to the player. Every round ends in a play of the game that replays a nice moment in the game when someone played well, then a screen of several notable achievements by players from both teams are shown in the form of cards. Say if there were some supports and tanks that did a lot for their team, then how much they healed and prevented damage would be highlight under their names. Everyone on both sides votes on who they think contributed the most and you end up with a lot of situations where a player on the loosing team will be the highest voted player, or there will be a single player that most people on both teams will agree did awesome. Better still, the feedback actually has a very concrete benefit, the match making system tries to match you with players you've previously voted for and those who were in matches that you liked. The end result is that the game feels much less adversarial, which is kind of weird for a competitive game.
Even the little details retain the same kind of feel. The kill feed is off by default and many players choose to keep it that way. Kills aren't prominently tracked or displayed, sometimes as a side note you'll see how many last hits you did, but for offense classes assists are the primary stat tracked, not kills. The final seconds of every post match summary is finished off with your own personal stats and achievements for the match such as whether you broke your own personal best or how close you were to breaking one, what your contributions to the game were organized by how your class plays (so supports will track things like how much damage you healed, or how long you kept buffs up, etc).
So if you're on the fence because of worries about slower reflexes and worse aim, or how competitive and hostile FPSes can be, don't worry: Overwatch has your back.