You have the general gist of it. logistics was the key to Ottoman and Safavid armies. Unfortunately I am not well versed in Mughal history so I will refrain from referencing them. If I had to pick the most important parts of the Ottoman military it would be: Janissary infantry, their effective use of artillery, and their cavalry. Gunpowder weapons, specifically cannons, offered a solution to a major Ottoman problem of the 15th and 16th centuries: how do we destroy heavy fortifications? The Ottomans were famous for their use of artillery, as well as the size that their cannons could be. But, in my opinion this is an often over emphasized aspect of Ottoman artillery, their real specialty, and what really helped defeat European and Safavid armies, was their ability to produce large amounts of light-weight, easily transportable cannons.

"But their field guns helped the Ottoman army to defeat their only serious Moslem rivals, the Safavid Shah's cavalry, at the battle of Chaldiran in 1514. (The Turks carried their comparatively light field guns in wagons and deployed them for battle in the gaps when the wagons were hitched together to make a lager for protection against cavalry attacks)."- The Age of Gunpowder Empire: 1450-1800, William McNeill, page 33

The main Ottoman advantage in battle was their cavalry, which used no gunpowder weapons at all. supported by the Elite Janissary Corps, the Ottoman military was a force to be reckoned with. It is also important to note that all three empire, specifically the Ottomans and Mughals, were fabulously wealthy, this allowed them to maintain large standing armies, as well as to create large amounts of gunpowder weapons.

But how did the Ottomans move their large armies so far, and so quickly? especially with all the supplies that would be needed? Well, two things helped the Ottomans move. First, good road systems throughout the more secure parts of the empire allowed easy transport, and a large navy allowed for transport by sea if applicable. Second, is the creation of imperial depots throughout the empire, unfortunately we do not know where most of these depots were because the Ottomans were unwilling to write about them for counter-espionage purposes, and they often moved around. The depots allowed the army to leave, say a staging city like Edirne (adrianople), without the need to bring all the supplies that would be necessary, they could pick them up on the way.

The Safavids had a more difficult task, unfortunately for them most of their territory is rather mountainous, and building an adequate road system that could easily transport artillery was difficult. They mostly relied on Calvary, and some historians (like the aforementioned McNeill) don't think they belong in the gunpowder empires conversation. I happen to disagree with this because they did adapt their military for gunpowder usage, even if most of it was still cavalry based.

To answer your question in the simplest way, I believe much of the uniqueness of the Islamic gunpowder empires stems from European interest in the Ottoman empire. Their ability to sweep aside European fleets and fortresses with mass use of artillery, as well as their ability to defeat non gunpowder states, or states that used gunpowder in a limited manner, garners a lot of attention from even Western sources. When you take a deeper look at it the Ottoman logistical ability to move massive amounts of troops and artillery gives them a clear advantage in most battles.

There are some good books on both Ottoman and Safavid use of Gun powder

Probably my favorite is this book which is an in depth look at the Ottoman military.

this is a good book on the Ottoman industrial complex and their ability, both from a military and economic point of view, to create large amounts of gunpowder weaponry. It also dispels the myth that the Ottomans were always lagging behind European armies technologically.

and while we are talking about that myth here is an article about the use of volley fire by the Ottomans in 1605, that predates European use (with the possible exception of the Dutch) of the volley fire tactic.

here is a book on the Safavid military, from whom I believe is by far and away the best Safavid historian.

I hope this helps to answer your question, if anything is unclear, or you feel I did not answer your question properly let me know.