I've written about it few times before, and basically, your hunch is right, the Ottomans nor the Mamluks never stopped the trade, nor was Constantinople that important for spices in the first place as amounts of spice Venice imported was much more from Egypt and Levant (and Venice had huge trade with Egypt), nor does the fall even matter as the Portuguese started going down Africa even before 1453.

To put it simply, the Iberians just wanted to get to the source of spices and buy them cheap and sell expensive in Europe, Ottoman actions don't really matter in this decision process.

My previous answers, the best is probably the first:

• r/badhistory - 'Spice must flow' a.k.a 'Ottomans stopped the spice trade and started Age of Discovery' myth

also:

• How much did the Fall of Constantinople influence the Age of Discovery?
• Did the fall of Constantinople stop the spice trade and cause the Age of Discoveries? Answer: no and no.
• To what degree was the Ottoman conquest of Constantinople responsible for Iberian interest in a new route to Asia? With data about Venetian spice imports prior and after 1453

You are right to question it. The story makes no sense.

1. Genoa and Venice were already trading in Egypt and Syria in the 1300s and early 1400s.^1 As you already seem to know, these areas were controlled by the Mamluks, and so were outside the Ottoman state.

We know from letters that Genoese and Venetian ambassadors continued to negotiate trade with Mamluk officials after 1453. One letter from 1506 mentions the new factor of Portuguese trade directly with India, and what steps the Mamluks and Genoese could take to protect their market share.^2

2. Genoa and Venice were already trading with the Ottoman empire in the 1300s and early 1400s. There are several examples of Genoese trading with independent cities on the Aegean coast of Anatolia, and the Ottomans then conquering those cities. In those cases, Ottomans continued the status quo trade policy, and trade carried on.

In the case of Constantinople, it does appear that volume of trade with Genoa diminished over the late 15th century. But, the reasons for that decline remain ambiguous, and it does not seem that Ottomans suddenly became hostile to trade.^3

3. The Portuguese had begun exploring the Atlantic coast of Africa in the 1430s, about two decades before the fall of Constantinople. However, Iberian discovery or re-discovery of atlantic islands like the Azores, Canaries and Madeira started back in the 1370s. In any case, Henry the Navigator was not looking for a route to India when he sponsored the earliest voyages.

In 1415, a Portuguese army captured the Moroccan city of Ceuta. The reasons for this are debated, but a plausible cause is that king Joao I believed it would give Portugal better access to the Trans-Saharan trade in gold. Another plausible explanation is that it was a step in Portugal's rivalry with Castile. Castile had begun the conquest of the Canary islands in 1402, and attacked the Moroccan city of Tetouan in 1400. So, a Portuguese attack on Ceuta could have been a move to make sure Portugal was not frozen out by Castillian and Aragonese conquests in North Africa.^4

In any case, this provocative move put Portugal in conflict with Morocco, and holding on to Ceuta became a drain on the treasury.

So, it has been suggested that prince Henry's sponsorship of voyages to settle Madeira and the Azores in the 1420s were counter-moves to the earlier Castillian conquest of the Canaries.

It has been suggested that expeditions down the Atlantic coast of Africa into what is now Mauritania and Senegambia were with the intent to "turn the muslim flank" and make contact with peoples who could be converted to Christianity and made into allies against Morocco.^5 Other explanations suggest that prince Henry's motives were largely commercial, and that the Portuguese anticipated finding the goldfields that were the source of the Trans-Saharan gold trade, and establishing profitable trade at the source.

In any case, from 1420 to 1488 successive Portuguese expeditions reached Senegal, then Sierra Leone, then the Gulf of Guinea, then the mouth of the Congo, then finally the Cape of Good Hope. The point of these expeditions were to explore and make trade contact with new markets in Africa. These expeditions were each successful and profitable to encourage successive, more ambitious follow-on voyages.

It's really only after Portuguese reach Benin in late 1470s that the purpose of the voyages then becomes discovering a route to India. Diogo Cao's voyage in 1482 becomes the first royally funded voyage (as opposed to prior merchant funded, royally sponsored voyages), with the intent to circumnavigate Africa. He failed, but did make contact with the Kongo Kingdom. Bartholomew Diaz pushed further in 1488 and reached the Cape of Good Hope. Vasco Da Gama could be reassured in 1498 that even if he failed to reach India, he could still expect to make a less profitable but successful trading trip along the African coast, because previous expeditions had demonstrated the financial viability of such voyages.^6

4. Now, there was a crisis of trade in the Levant for about 50 years after the fall of Acre to the Malmluks in 1291. Eliyu Ashtor covers this in chapter 1 of his book Levant Trade in the Middle Ages. This event may be the origin of the mistaken idea that fall of Constantinople led to trade crisis because of Muslim intransigence.

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1 Levant Trade in the Middle Ages by Eliyu Ashtor, particularly chapter 7 "Medieval Levant Trade at it's Height (1453-1498)"

2 Ottoman Seapower and Levantine Diplomacy in the Age of Discovery by Palmira Johnson Brumett. pp 197, note 51.

3 European and Islamic Trade in the Early Ottoman State by Kate Fleet. pp 124-125.

4 Foundations of the Portuguese Empire by Bailey Diffie and George Winius. pp 49

5 Africa and Africans in the Making of the Atlantic World, 1400-1680 by John Thornton. pp 24-25

6 Thornton pp 31