It's true that the Ottomans relied very heavily on the support of the army (read: Janissary Corps) but it worked out well for them at first. Though there WERE hints of future problems as early as the 1400s, the Ottomans had such a rash of strong leaders that the early sultans could appease them and still basically do as they wished. But that also leads to another problem. The empire was so enormous that people could do things out of sight of the central government for years and the borders were so vast you might have to let internal problems be for a decade while you dealt with the Persians, or the Austrians, or Poland-Lithuania, or the Russians, or the Spaniards etc.
That seems to be the worst problem (underlined). Everywhere the sultan looks is something needing his attention and the state's resources - and the borders (the actual line next to enemy territory) are ill positioned for Constantinople's eye to be on more than one at a time.
It would be interesting what an Ottoman domain the size of France, rather than an enormous pan-Sunni empire could accomplish in building a state apparatus. Compare the Russian empire, but Russia had railroads and Christianity, which is more hierarchical until you hit the afterlife.
How does Russia in this period (first half of the 17th century) compare size-wise to the Ottomans?
It was pretty huge even before railroads.
ED: What would have happened to the Janissaries? That's a good question. My guess is they'd be broken up and integrated into the larger structure. But that would have required the state to build a parallel army of some size first, or as happened in OTL get the support of the other branch of the army, the Sipahis.
Which brings up its own set of charming problems, unfortunately.
I think this is where the term overextension comes in. Between external threats and internal ineffeciencies, there's only so much the state can do - and so much that the state has to do.