Captain America: Civil War is the first time we ever see Tony's mom. Apparently he loved her very much. That's interesting because based on his past characterization I had always assumed he had no mother and was grown in a vat.
Still, Tony Stark had a mother. She died. For three Tony Stark movies and two avengers movies no one gave a damn about his mother or her death.
This movie, though, hinges on her.
Everything comes down to the fact that Tony will stop at nothing (not even killing a friend) to avenge his mother in a "Life for a life; I don't fucking care he was brainwashed," kind of way.
It is significant, I think, that after learning that the Winter Soldier killed both of his parents Tony never mentions his father.
In theory Tony got closure on his daddy issues in Iron Man II, but in practice he doesn't give a shit about his dad's murder but good fucking god is he pissed about his mom. This is mildly interesting because at the beginning of the movie he says his big regret was not telling his dad that he loved him.
Doesn't really matter though.
What matters is that Tony will stop at nothing to kill the longest held, arguably worst treated (mind control is a form of torture that, thankfully, doesn't exist in the real world), prisoner of war in history for something he isn't responsible for and a villain who has never even met Tony could figure that out.
All he had to do was put Tony on a collision course with Bucky and Tony would try to kill Steve and thus tear the avengers apart.
So, in a sense, that's why Tony is in the movie. No one else is enough of an asshole to really make it work. Especially since, as of that point in time, all of the original avengers had been mind controlled to one extent or another and thus understood how that interacted with responsibility and whatnot.
Natasha, for example, immediately put the breaks on Hawkeye's post-mind control self-inflicted guilt spiral because the killing he did while under mind control couldn't and shouldn't be blamed on him. It wasn't his fault.
Even Tony should get this because the reason he created Ultron and thus nearly killed most life on earth was due to having his head fucked with by Wanda.
But he's Tony.
So the villain needed to draw Tony in. I get that.
If you're going to use a hero to accomplish your villainy in the MCU your best bets are Tony Stark and his sidekick Rhodey. Unfortunately to get Rhodey you generally need to be in a position to manipulate the US military at pretty high levels. Also Rhodey is probably going to figure out eventually.
Tony, on the other hand, is a canon that's so loose it might fall off the damned boat (or spin around and take out the crew) and so he can be manipulated much more easily. Instead of trying to manipulate an organization (the US military) you've just got to manipulate one person.
Tony is the weak link, the low hanging fruit, and the easy mark.
It does, in fact, make sense for Tony to be in this movie. What doesn't make sense is how much he's in the movie.
Tony's only reason to be in the movie is because the villain wants him there. That's easily accomplished. The villain has information that Tony wants. He could have been dropping bread crumbs for Tony that would ultimately lead Tony into the final confrontation.
So we'd see the main action, cut to Tony finding another clue and getting closer to the truth, (which would be a very short scene) and then cut back to the main action.
It wouldn't be until the end, when Tony's search leads him to the same place as Steve and Bucky, that we'd understand why we've been having these brief cuts to Tony's investigation, and even then we wouldn't know for sure until, like in the actual move, the villain's plan kicks into final thing:
Tony gets to watch his parents being murdered by Bucky back when Bucky was brainwashed. Tony goes into murder-death-kill mode. Steve goes into "What the fuck man; he was brainwashed!"
So then the question becomes, why the hell was Tony in the movie as much as he was and in the way that he was?
Tony is more or less retired. He isn't even working with the fucking avengers when the movie starts and the role he seems to be filling until he gets a black eye is not his damned role. He stole it from a female character.
Maria Hill is the one who does the shit he's doing. Why did he steal her job? When did he steal her job? After the Ultron fiasco, why would anyone trust him enough to let him steal her job?
Even if he somehow manged this usurpation via the ancient rite of "I control your funding; do what I say" why is no one pissed off about this? Why is not a peep uttered?
One might believe that he could be called in for the big fight because he makes weapons of mass destruction the way most people produce carbon dioxide, and he incorporates them into his suits the way most people ... um, assimilate ingested nutrients?
But he wouldn't fucking be leading one team. There's no call for that. Probably Rhodey should be since he was the only character to put forth a cogent argument in favor of that side.* Failing that Hill could be in the lead but they'd probably have her do it from off-site since, while she definitely has the skills for field work, she seems to prefer doing things back at base.
There's a reason for him to be there at the end, we covered that, other than that pre-black eye he was stealing Maria Hill's job, and post black eye he was stealing Rhodey's. With one exception.
When Tony goes to recruit Spider-Man that's ... how the fuck did Tony know about Spider Man or know that Peter Parker was Spider-Man to begin with?
Totally serious here. It just drops out of nowhere as a complete non-sequitur and leaves us wondering, "What the fucking fuck was that?
The most logical explanation, if applying logic to this movie is itself logical, is that Jarvis somehow found out while he was off being half/dead semi-merged with the internet during Age of Ultron and passed that info onto Tony post resurrection (either as Jarvis pre-Vision creation, or as Vision post-movie.) However if that is the case then Vision should be the one recruiting Peter (possibly by proxy if it was thought Vision of the Red Skin would be too disconcerting for Aunt May.)
If Vision had done it then it might do a bit to explain why a team of "heroes" recruited a child soldier. Vision is, age-wise, an infant and thus might not understand why recruiting a child, someone older than him, would be a problem as compared with recruiting an adult.
But ... we've got no fucking clue how Tony knew and thus no way to make sense of it for in-universe reasons, and thus no idea if it actually made sense for Tony to be there.
Ok, so I think that covers everything
In the beginning the role filled by Tony should have been filled by Maria Hill.
When Tony recruits Peter there is no sense and reason has been banished. We can reach no conclusions.
After that what he does should have been done by Rhody until...
At the very end Tony actually belongs there because no other avengers are quite fucked up enough to be manipulated into doing exactly what villain wants and thus the villain wouldn't have chosen any of the other ones.
* If you missed said argument, don't feel bad. Maybe you just sneezed.
The bastards who made the movie didn't give Rhodey much in the way of lines or characterization so it's easy to miss pretty much anything he does other than get hurt and cause Tony great "I am a white man and my black friend is hurt" sympathetic pain.
Way more was devoted to Rhodey as a vehicle by which we can experience affluent white pain than Rhodey as someone capable of explaining the problems with Steve Rodgers' vision of the world.
Non-Wakandan black lives may matter, the movie sort of concedes, but a) only insofar as white people have feels about those lives and b) their lives mattering sure as hell doesn't mean their thoughts matter.
And, yes, that's really, really fucked up.