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 
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.
Dr Who's visual continuity on the other hand is pretty strong when bringing back old creatures. Daleks still look like they did in 1965, the Sontarians as they did in 1973, the Zygons as they did in 1975, etc. The only race that got a major change was the Cybermen which was explained by being an alternative universe version. Even they eventually cycled round to their original 1966 look by the end of Capaldis era.
I mean... not that much. Daleks have gone through three redesigns just since the show went back. Sontarans went from the world's most unconvincing rubber masks to makeup. And how many eyes do Silurians have-- two, or three?
The point is though that DW confirms again and again that how things looked is accurate. We see a 70s Cyberman helmet in 2005, before we see the Modern Cyberman. We've seen the Dalek redesigns happen, and past designs all the way back to the 60s reappear.
Sontarans are a clone race, so its not difficult to imagine changes happened to their process or gene template over time. Perhaps one day we'll see a mixed Sontaran fleet with the short stocky guys from Series 4 and the taller ones from Classic Who and Chibnal. Perhaps Silurians have different subspecies. Those are easy, one sentence explanations, that don't rely on scoffing at old SFX and going "well its broken already anyways".