Praxis was just the most obvious symptom--it wasn't the disease, it was a result of the disease. Chernobyl didn't bring down the Soviet Union, either, but it displayed for the world to see the weaknesses of the Soviet system.
I'm also not sure the economic disarray which eventually led to peace in the "real" universe necessarily precludes the Klingons from fighting a twenty year war with the Feds, either. If the Klingon Empire declined as the Soviet state did, the Federation and other powers probably aided the ailing nation with measures which were painful and prone to creating a lasting bad impression of the Feds with many of Klingon people. And even with the economy in tatters, the Klingons would still have been formidable--the machinery of the Klingon Defense Force, the thousands of starships and millions of soldiers, did not vanish with the economic contractions. At the same time, I imagine looking at a Klingon world in the YE timeline would yield a very depressing picture of abject poverty. (Of course, abject poverty in 24th century terms probably means they live merely slightly better than we are accustomed to.

)
Still, in the "real" universe, the Klingons eventually managed to return to their place as the no. 2 or 3 power in the AQ by nursing at the Federation teat instead of squandering the resources they had left with an unwinnable war of revenge, which led to a particularly neurotic inferiority complex amongst some leaders compelling them to oppose Federation policy and eventually attack the Federation in a totally self-destructive effort.
I view the relationship after the DW especially as similiar to that of the United States (Feds) and United Kingdom (Klings) after World War II. Sorry about your colonial empire, losers, America believes in freedom! Also culturally ignorant and overly aggressive anti-communism!
Btw: Praxis need not have been a moon of Kronos to have destroyed its ozone layer. If the somehow-superluminal phenomenon (I hate that) included a significant component of x/gamma rays, the intensity could still be enough to photodissociate the ozone, leading to ultraviolet poisoning of plant life and ecological catastrophe, etc. Although you'd think in the 23d century ozone wouldn't be that hard to replace.