Really, that's what they should have done. In TnT they had a chance to solve it, once and for all. They could have just simply ignored the differences and had everyone act as if the Klingons looked the same as modern-Trek ones or, hell, had an introduction by Dorn at the beginning of the episode saying something like this:
"Welcome, to Star Trek: Deep Space Nine's episode, "Trials and Tribulaitons." Tonight you're going to see your favorite Star Trek Crew visit the classing original series episode "The Trouble With Tribbles." This landmark episode is a cherished favorite with fans of the series, casual fans and Trekkies alike. When the original series was made in the 1960s they worked on budget and time constraints that didn't allow for the make-up we see and enjoy on television today. This lead to many of the aliens on the show to simply looking human, including the Klingons. It wasn't until the Motion Picture in 1978 that Trek's producers were able to truly bring Klingons to life and give them a look of their own. Tonight we acknowledge these differences but also say that the way Klingons look today are how they always looked, so our crew will not notice these obvious differences. We ask you to imagine along with us that these Klingons always had the fierce warrior look we enjoy and love today. Q'Apala! And enjoy the show!"
They could have just had the crew ignore it. Pretend nothing is different. But they HAD to address it, and Enterprise just HAD to try and to explain it. And that created even MORE problems and silliness. The explanation they came up with, yeah, I guess "mostly" works. Still, it would have been far better to never address the difference to begin with. Accept it for being what it is, just the differences of the budgets of the times the series were made in.