I am the kind of person who enjoys "big weird" scifi like Stanisław Lem. Stories about trying to relate to and find common ground with something so alien that the prospect of even understanding is basically hopeless. Star Trek usually doesn't do stories that, which makes sense as it often uses alien races as allegories or stand-ins for real-world human relations.
That said- I thought those early Klingons were super weird and scary because they were just so alien. It really made sense thinking about how it took a century before they could get to the events of Star Trek VI, and it made the Khittomer accords feel like so much more of an accomplishment. Like- you made a treaty with WHAT?
And just aesthetically their ships and armor looked like something out of HP Lovecraft or HR Geiger:
This is not to say I dislike how Klingons were portrayed previously, kinda like Mongols in TOS or Vikings in DS9, just that they never felt scary to me. They never felt like warriors. I was never afraid for the gallant crew of the Enterprise D (a science and exploration vessel) going into battle against Klingons. But I really enjoyed the alien-ness Disco tried to go with. Anyone else with me?
EDIT: PEOPLE I SAID WHO'S WITH ME NOT WHO ISN'T CM'ON 
For me, having them look like TNG Klingons doesn't even solve the problem because ENT had implied that shouldn't happen until the TOS movie era. They could have rendered explicit the implication that not every Klingon was infected by the virus, but that still doesn't support making the Klingons look how they did in s1 DIS.
That Enterprise arc was clearly intended to apply a (totally unnecessary) in-universe explanation for why Kirk's adversaries were just guys in vaguely asian facial makeup, but there's no reason we have to extrapolate that the Augment virus was a widespread and incurable until the late 23rd century. It could easily have been a relative blip on the radar; aggressively quarantined and/or cured much earlier than anticipated.
The idea that they also needed to make an explicit reference to the augment virus being cured, or explicitly point out "hey, these guys would look less different if they weren't shaving their heads!" strikes me as absurd. These are not difficult conclusions to reach for someone motivated to find them, and there were people mentioning those possibilities pretty much immediately after the first Discovery trailer dropped.
All that needed to be maintained was that the Klingons we see Kirk face in TOS were all afflicted by the virus - while it's still reasonable to assume that, the presence of these hitherto unseen 3rd variant of Klingon complicates instead of simplifies, which is what ENT's arc did. Now what, it's ANOTHER coincidence that THESE klingons are even ridgier than we've seen before, but the other ones are still out there? To borrow your parlance, the Discovery redesign was intended to overwrite and replace what came before, because apparently Star Trek, unlike every other fantasy and science fiction thing I like, is Forbidden from being treated like a secondary world that should have its own internal consistency.
I was completely content to accept it was a coincidence that Kirk only saw augment virus-impacted Klingons in TOS, just like how ST Picard ended up establishing for Romulans (northern vs southern to explain the v shape bone ridge they had through TNG-ENT).
How many other Science Fiction properties out there sprung out of a low budget TV show from the 60s but are still producing content in the same continuity without some kind of explicit reboot?
Star Wars is the classic comparison in all sorts of ways, and for better or worse Star Wars avoids this problem entirely by 1) having a much higher budget relative to the number of sets and costumes required for it's initial installment, 2) having picked an aesthetic that is crude, gritty, and seemingly practical which escapes looking dated many years after the fact, and 3) not being set in our future where the advances of modern tech make obviously retro elements look ridiculous.
There is Doctor Who, and that's it.
Actually, DW is a good example, because the continuity of that show is a mess, and very intentionally so. That show thrives in its inconsistencies. There are three different explanations for why the Doctor can change faces when he dies, for example. And each one contradicts the others. There is also no beta canon, every tie in is considered canon. So the doctor has officially met Batman, Gandalf and Picard. That's canon.
In the end IP is a playground and continuity should enhance story. Nobody gains anything from lore for the sense of lore. What does the Klingons always looking a certain way say? Not all that much. It's a nice to have, because it allows you to recognise them quickly and make connections. But if the look is constricting for the creative team, then they should be able to change it.