I'm not sure I buy into the whole 'sick man of Europe' thing to the same degree that others might. Without the war, the Ottomans can make some extra cash, and perhaps enact some reforms, and perhaps linger on for a good long time. Not sure that Arab nationalism would be quelled permanently, but it might be placated for a good long time.
Let us say the Russian Revolution still happens - and that the Bolsheviks still take power and win the Civil War.
It's not a guarantee in a timeline like this, but still...fairly likely.
Well, if it does, imagine how this affects British and even French geostrategy in the 1920's and 1930's. If the thought of the Tsar getting his mitts on the Turkish Straits made Victorian policymakers queasy, the prospect of the Bolsheviks getting hold of it will give their postwar successors in Whitehall strokes. Suddenly, the Ottoman Empire looks much more attractive as a counterweight for containment of the Soviets. And that means keeping it as strong as possible.