It's not awkward. Chapters titled with the name of the person who's experiencing it. The chapter from Kirk's POV is called Chapter 1: Kirk. Then Chapter 2: Spock. Chapter 3: Scotty. Et cetera.
Actually that does strike me as a little awkward. I mean, it's been done, but it's a little... self-conscious, it seems.
The thing is, just talking about it as a device out of context, it can't help but feel contrived. It's the wrong way of approaching the question. You don't start with a device and then concoct an excuse for using it. You start with the story and use whatever storytelling mode works for it. If the story needs
to be told in alternating first-person chapters headed by the characters' names, then it wouldn't feel awkward (probably). Otherwise, it just feels like an imposition.
Sounds like you're agreeing with me. It just seems awkward to you because it's taken as a device out of context. By your own admission, if there was a story that suited the "Chapter 1: Kirk" style, then it would be perfectly reasonable and not awkward at all.