They couldn't beam them up because they were moving too fast. That's why they beamed her down in the first place. And yes would have worked better if there was some reason, any reason why it had to be her beamed to that moving platform. A reason they were not able to beam down a security guy.
why couldn't Spock send down some security guys too? He was the acting captain in that moment and he abandoned the ship. Same for the first movie when he went looking for his parents in the imploding planet of vulcan because he wanted to save them.
A personal reason - she's his girlfriend so she won't let others do the job that she can do- would be, alone, a legit reason for her to go and nothing new in terms of how these characters act.
Kirk went after Harrison personally because he killed Pike. Technically he didn't have to go he could've send an away team of security officers only and never have to leave the captain's chair to Sulu. That he took Uhura with them makes sense as she was the one who spoke klingon but Spock for example was replaceable, what he could do aside from telling him that the statistical likelihood of the villain trying to kill them was around 90%? ^
Another reason, though, for her to go could be that they needed someone who could be able to
stop Spock before he killed Khan and he'd probably listen to her a tiny bit more.
Okay, so she was courageous to be willing to "talk" to Klingons, and of course things went south. My issue is that that "talk" didn't do anything. If you're satisfied with it, okay, but I can't be.
why the plan had to work for the scene to be important and relevant for her character?
The scene's purpose is to show her competence and that she is so brave and courageous to be willing to risk her own life just to give to them all a chance to survive well knowing that she could fail and that really the chance that the klingons would listen to her were very little. But she TRIED well knowing the risk.
Even if she just made them gain some time it's still something.
The scene also showed that while S/U are in love and have a relationship (something touched on in a previous scene) this doesn't affect their job. Kirk doesn't want Uhura to go outside and he wants to stop her but Spock recognizes the logic in her plan and the fact they don't have other options and he believes in her and her ability to handle the situation on her own.
Then later when they're in the middle of the fight you don't see them going after their boyfriend/girlfriend uncaring of other officers that could be in danger in that moment. When Kirk is down and maybe injured their priority is to help their captain and they drag him out of the line of fire.
