Can you recall any prior series of Star Trek where the Klingons were wiped out in two alternate universes and (mostly) the main one?
Obviously not that specific sequence of events, no. However, in addition to seeing Praxis explode in the 23rd century, in the 24th we saw the Klingon civil war, the underhanded instalment of an illegitimate leader, and various ugly internal power struggles, which all culminated in this memorable take-down by Ezri Dax.
If I can watch a likeable character like her calmly declare that "the Klingon Empire is dying and deserves to die" and not think the Berman era writers hated Klingons, I really don't think there's anything you can point at to make me think the Kurtzman era does.
It just seems so over the top to assign these underhanded motives to the writers on so little justification. It's clear to me that they're treating the Klingons as an important part of the mythos. If nothing else, Jay-Den is easily my favourite of the Academy cadets. I'm just not seeing what you see here.