In my head canon, the Klingon's spent the decades after the outbreak of the Augment virus trying to genetically engineer their way back to authentic Klingon and DNA. Much of it was superficial, or surgical and for looks, but largely failed. Around the 2240s they attempted to go even further and enhance their species DNA with new traits (long head, claws, purple skin) and parts the old Klingon look, and that is how we got the Discovery Klingons. The Discovery design was an exaggerated attempt at restoring Klingon traits (and then some) as an expression of Klingon cultural and racial distinctiveness.
T'Kuvma's "Remain Klingon" bit was in part literal - a societal humiliation of the Klingons by the fact that human Augment DNA had thus far proven impossible to expunge from the Klingon genome and the Discovery Klingons were the result of an aggressive attempt to undo that humiliation. And true, they certainly didn't look human anymore, but also not really Klingon. The 2250s attempt turned out to be unstable and ultimately a dead end. Do they were stuck with mostly the TOS look through the 2260s as they refined their genetic engineering technology.
A scientific breakthrough late 2260s lead to the TMP Klingons, which, with further development, fully repaired the Klingon genome, free of Human Augment influence, starting in the late 2270s. The Empire was gradually treated over the 2280s and early 2290s. By 2300, nearly no TOS Klingons remained.
The reason I like this in my head cannon is that it provides an effective augur for a century of Klingon-Federation (particularly Klingon-Human) hostility. It wasn't just a bad first contact, or some military exchange or border dispute. The Augment virus removing Klingon features was, from the Klingon's perspective, unprecedented societal assault... even a form of conquest upon them... that was a multi-generational humiliation that fathers would pass to sons. They no longer even had the "look of Kahless". Peace between the Klingons and the Federation, though activated by a changed political situation in the Quadrant, was made possible because that societal humiliation had receded as the legacy of the Augment virus was expunged form their DNA. They would "remain Klingon" after all, even though it took another 30 years of technological advancement after T'Kuvma's attempts. The 100 years of affliction by the Augment virus's genetic legacy, however, made peace impossible before hand.
In this way, I think my head canon turns the horrendous Discovery Klingon design into something that both makes sense, fits with canon and makes it occupy a meaningful place in Klingon-Federation history. Ultimately it's exaggerated look is intentional - a exaggerated attempt to "Remain Klingon" and not become indistinguishable from humans, and makes the Klingon's century of hostility towards the UFP understandable and nuanced. Look at it from the Klingon's perspective: their empire long predates the UFP, it is more powerful than the early Coalition of Planets and UFP, but then humans infect them with a virus that makes them not look Klingon anymore, and within a few decades, the UFP spreads all around them like a virus. From the Klingon perspective, that would be an attack on their very being. Which makes a genetic engineering attempt like the Discovery Klingon's logical, and T'Kuvma's "remain Klingon" rallying line, and the TOS/TMP/TNG designs all being co-hesive, and the 100 years of conflict.