One of the primary motivations for the European “Age of Discovery” was finding a trade route to India and China that bypassed the Ottoman Empire. The Ottomans themselves, of course, didn’t have that problem.

No you're mostly correct. The Spanish and Portuguese that colonized South and Central America were both apart of the Catholic church. And would do many things, e.g colonize and enslave a continent, in claims of doing so for the church and spreading Christianity. As you are probably well aware of the pilgrim story of Massachusetts, religion was apart of the colonization of north America too, specifically Christianity as well. England, of course, was aligned with the church of England. And France with the Catholic church.

Islamadom at the time was focused on India.

Islam was busy spreading to the Pacific through Persia, India, and to Indonesia (the most populous Moslem country in the world.)

In addition, they did not have a significant Atlantic Ocean sailing/naval presence at the time, as they had recently been pushed out of the Iberian peninsula (Spain and Portugal.) While Arab and Moslem traders and sailors were well familiar with coastal sailing in the Atlantic, they had no Atlantic-facing major port trading city with which to launch transoceanic voyages.