@BellaGerant, so what your saying is the Ottomans were simply in the right place at the right time? That seems oversimplistic; I mean if that was the case, why did an Ottoman Empire rise instead of say, a Karamanid or Germiyanid Empire instead?
No, I literally said "It's hard to pin down singular causes for the Ottoman Empire's rise" and my focus on the geopolitical situation of the Balkans in the 14th century wasn't meant "to downplay the actual military and administrative talent that kept the Empire growing for 3 centuries." There were many different factors, military, bureaucratic, economic, etc. that went into the success of the Ottoman Empire, one of which is the geopolitical aspect that I mentioned previously.
Anyways, if we're still focusing on the geopolitical aspect, the Ottomans being on the westernmost part of Anatolia mean they were right next to the core of by-then abject and crumbling Byzantine Empire. This let them swallow up Bithynia with relative ease while the Byzantines were beset on three sides (Serbs to the west, Bulgarians to the north, Ottomans to the east), which forced the Byzantines to make concessions to the Ottomans that helped facilitate the latter's initial expansion into Europe (Gallipoli under Orhan I), gaining them valuable loot and resources for war in all directions. This allowed them a position to take Eastern Thrace, which further cut the Byzantines off from the rest of their holdings and allies while allowing the Ottomans to exploit Byzantine infrastructure for their own administration. Being on Anatolia's western coast also helped the Ottomans secure all routes from Anatolia into an increasingly unstable Balkans, which prevented other Turkish beyliks from raiding and invading the Balkans in the same fashion.
Those are things that the Ottomans had over the Karamanids and Germinyanids, which does factor into the equation.
At the same time, I'll again say that I don't think that that was the only reason; it certainly helped but such events rely not only on position but on the gumption, wherewithal, and, dare I say, talent to identify the opportunities afforded by one's position and to pursue the decisions that would allow the exploitation of said opportunities, the Ottomans having all of the above during their establishing period.
Power vacuum was, indeed, there but to become a major expansionist power you need to have an instrument allowing you to fill that vacuum. For the Ottomans such an instrument was a regular infantry they created, janissary. While being just a component of their whole military system it was giving a winning edge against the opponents who did not have anything of the kind.
Again, in my defense, I did say that it wasn't the only reason and that what I said wasn't meant "to downplay the actual military and administrative talent that kept the Empire growing for 3 centuries."
Simply that that was a contributing factor that allowed the Ottoman's effective institutions and leadership to excel past all expectation in their meteoric rise, as opposed to pure luck as posited before my post. Was it the only thing, no, that's an oversimplification. But was it an advantage they had over the other beyliks? That's a bit easier to justify, I'd say.
If there was no power vacuum, the Ottomans wouldn't have seen as much success as they did OTL is something I think is reasonable enough to state. In that case, it does relate to OP's question, even if it doesn't act as the complete answer.
There is no doubt that the events of 1204 were a massive blow to the Romans, but I don't think it was inevitable that they would collapse. The collapse of the Sultanate of Rum gave them a new opportunity. In 1300 they were still in a decent position. Unfortunately they had poor leadership over the next century and that was really when things were lost for good.
Well, the Empire of Nicaea had only retaken Constantinople in 1261 and they had lost almost all of their Anatolian holdings between then and 1300. To say that the Byzantines were in decent shape in 1300 is a bit generous, all things considered, seeing as their navy had been dismantled due to costs, they were still at odds with Genoa and Venice (the latter of whom they were at war with in 1300), and the Turks were still pushing into the eastern border, made especially weak with the transfer of troops from the east to fight in the west against Serbs, Latins, and Bulgarians.
The collapse of the Sultanate of Rum afforded them little in opportunities as the Serbian and Bulgarian Empires kept pressing into Byzantine lands and forcing their attention westward, that being one of the reasons the Ottomans were able to fill the power vacuum in western Anatolia in the first place instead of being stomped out in their infancy.
Was Byzantine collapse guaranteed? Perhaps not, but it was certainly more likely than not even in 1300.