So Mom and Grandpa, and then even Dad, all worked in Starfleet, and that's how David Marcus hated Starfleet?
Well in this new universe, Carol must've followed her dad into Starfleet. In the Prime Universe, she apparently didn't, or at least didn't make a career out of Starfleet. Poor David probably won't even be around in the new universe to have an opinion about Starfleet one way or the other, as it looks like Kirk and Carol don't get together.
As far as David's attitude towards Starfleet in the Prime Universe goes, if Carol had been in it at all (no way of knowing), then she left for some reason that may have tainted his opinion of the institution. Maybe her opinion soured. One way or another, why can't David hate (or at least be skeptical of) Starfleet?