4 movies, so what can you show me that makes the audience want to care or feel for Rhodey. All I've been told is he's Tony's BFF and that's enough to make people care. That's very shallow. Vision is asked to shoot him down but on a story and thematic level was that correct? Why didn't the movie follow-up with Vision after? This is what separates, to me, good blockbusters from those that are just going through the motions. We know why Vision shoots down Rhodey but is it any good?
Wow...so you were being serious. OK, lets go through the movies. You want to know how the movies established Rhodey as an important person to Tony, right? So, just some small examples from each film.
Iron Man: Rhodey is who Tony calls when he's in trouble. Aside from Pepper, he is the first one Tony reveals his identity to. The relationship is shown to be more than just acquaintances or business connections through little details like the custom ringtones, the banter, and Rhodes' access to Tony's house.
Iron Man 2: Well, Rhodey and Tony are built up throughout this movie. From the senate committee, where Rhodey apologizes for being there, then does his best to protect Tony. Then you have the scene where Rhodey sees that Tony is dying, where he reaches out "You don't have to do this lone gunslinger act." Which leads to Rhodey trying to step in and put an end to Tony's self-destructive behavior at the party. Then, of course, there's the end, with the whole struggle between the two.
Iron Man 3: I'm not even going to go into this, as it's pure buddy partnership whenever they're on the screen together. Heck, by this time, Rhodey is so firmly established as Tony's closest friend, it's ridiculous.
Ultron: Rhodey doesn't get too much screentime here, but the banter between the two is still present. Furthermore, when the two are in combat together, it so clearly flows, reinforcing the relationship between them.
Civil War: The big thing here is that Pepper and Tony are on the rocks. This defines Rhodey as Tony's partner. There is no other way to look at it. The previous movies show that where Tony goes, Rhodey supports, and this one has it going the other way. These are long-time friends, a relationship that is older than ANY other in the movie save for one. So, yeah, when Rhodey is blasted out of the sky, it has weight.
As for a thematic reason...are you serious? Did you not get that the theme of the entire movie centered on the consequences of the Avengers' actions? That all they were talking about over and over and over was collateral damage? Billions of dollars in damage, loss of life, etc. Tony is only an observer to the personal price when he speaks with the woman at the elevator. There are warnings again and again and again that if they continue down this path, a huge price will be paid. Rhodey getting shot down makes Tony no longer just an observer, it brings it home.
As for why it was Vision that took the shot, that's made obvious as well. He delivers one of the clearest warnings on the chosen course of action, AND he repeats that warning just before the shot. The movie hammers the viewer over the head with it, just in case they didn't catch it the first time. "Hey, remember this? You're going to need this fresh in your mind to fully grasp what we're doing here!" Movies do this all the time, it's part of good storytelling.
So, yeah, thematically, it fits.
edit: You asked why we don't follow up with Vision...the melancholy shot of him at the chess set in the empty room says it all. It shows how he was affected by what happened and how his world was effected. Don't need any more than that because, well, it's not his story.
