I'd probably find it a lot easier to rationalize the 'Discovery' production design being modernized, as
nobody was ever going to make a show with sets and uniforms replicating those seen in the sixties.

But it's actually the Klingon redesign (so closely aping that seen in the Kelvin films) that I find a little harder to take. I'm not usually someone who quibbles over these kinds of pointless details too much, life's too short for that.

But I can see myself finding accepting that change to be
just a little bit too far in my mind, and I'll probably end up seeing 'Discovery' as a prequel to the Kelvin films, rather than a prequel to TOS, just because anything else would probably bug me too much in the long run.
That's not to mean I'd have expected them to use bare-headed Klingons with racist blackface and Fu-Manchu beards.

But like in 'Enterprise' before it, something in the vein of the TNG Klingons or even the subtlety different TMP Klingons with their spiney-spines might have been acceptable. Those 'Discovery' Klingons are just too much like the ones in the Kelvin-verse for me to overlook it. (Although, funnily enough, both the 'Discovery' and 'Kelvin' Klingons seem to bare some similarity to the pre-evolved Worf from TNG episode "Genesis".) Are they meant simply to be
yet another strain/race of Klingons we've never saw before?