I won't even discuss the TOS part of your post, the fact is that in the Kelvin Timeline her only appearance was off screen and she was a nurse. Why is that sexist? Is being a nurse an inferior profession? A friend of mine is a nurse, and she would strongly disagree!
Chapel's 'fate' is not overtly sexist (although there are sexist undertones). In the Kelvin timeline anything goes so, as you say, why shouldn't she end up choosing nursing as a career path in favour of bio-research? I also apologise to any nurses out there if it comes across that I'm belittling their profession; my principal issue is from a character writing perspective i.e. that if you make Chapel a nurse again or even an MD, she has to compete with McCoy for air time and she's always going to lose. To gain credibility as a character, she has to be given a niche of her own where she can complement McCoy, and fancying Spock is already taken, if you can call that a niche... So she can be a xenobiologist, exopaleobiologist, a xenoarchaeologist, a bio-mechanical researcher, a geneticist, or even just a tribble expert. She could have made some contribution to complement McCoy's 'old country doctor' with a bit of imagination.
However, I did find it disrespectful for her to be cast aside from the franchise with a tongue-in-cheek line, since Majel Barrett had been gracious enough to lend her voice to the ship's computer and had recently died. Compare that to the reverential treatment that Nimoy's character deservedly received following his passing. It may not be sexist or it may be, but I found it distasteful either way.
Behind the scenes politics had led Barrett to step away as a member of the main TOS ensemble from STII onward until her return in TNG but her contribution to the franchise as a whole was considerable. I just felt that the writers could and should have shown the actress some respect by showing her character some respect (including making her less boring). They also left out Number One in ST09, despite a niche that she could have filled.
Carol, as a character, is in a similar position to Chapel in that, as a physicist, she is competing with Spock's niche. That leaves her romancing Kirk as a niche, which is pretty thin and is also straying into Uhura's territory. There's only so much romance one movie can take! They might bring her back with a toddler in tow in the next movie, neatly bypassing the romance, to give Kirk something new to emote about. I would like to see Carol return in some form though.
I'm also unsure why they didn't just bring in Janice Rand as a redshirt security guard. I'm not saying I'm anxious for Rand to die but I also can't see why they couldn't use such a simple way to update the character for the modern audience. It's not as if they'd be turning her into NuStarbuck :-p
