The only one I've really tried is Overwatch and personally I kind of like it.
Though I never played much of TF2, I know what you mean.
As someone who generally don't like MOBA's and I have found myself to quite enjoy Overwatch, despite the fact that I find it lacking in the map/game mode department. (I promise, I won't complain about the god damn egypt map with the bloody chokepoints that make me want to strangle someone...whoops!)
While the experience you posted in the purdy picture certainly can be true. I guess it really depends on how well you can handle it. I'm generally carefree in games. "So I didn't get the best team and a lot of stupid stuff is happening, eh whatever. I've done plenty of stupid stuff myself" is my general attitude about it.
I find this type of game (based on my impression of Overwatch at least). To kinda be a step in the right direction. It's a bit more creative. There's several playstyles depending on which character you pick. Lack of regenerating health is nice. And the fact that there are health packs on the map as well as having healing classes mean that you're not completely fucked if you don't stay around the healer all the time.
I think you can kinda throw Dirty Bomb into the lot with the rest of these. Several different characters/classes and so on.
In general though, I like this "evolution", simply because it puts more alternatives for different types of games out there. And the fact that big developers are behind quite a few of these is something that should interest the whole "AMG TEH GAEM HAS 2 B ESPORTZ!" crowd. It brings the focus back to the FPS genre that seems to have pretty much stagnated in the competitive scene. Go on, name three FPS games that are played in major tournaments and such. (Excluding CS:GO and any silly console stuff. Peasant contests do not interest us, right?)
I guess one of the main worries fans of the more AFPS genre have about these games is that they will be "Consolized" slowed down and so on. Which surely can be justified but also mended with the whole character system. The character I mainly play in Overwatch is Tracer because of the fast-paced nature of the character which makes it more fun to me. While the slower ones are simply not as interesting.
..No idea what my main point or anything is here but hey, it helped me pass some time