The premise of your question is not good. there were many voyages prior to the Renaissance. Hannibal the Navigator made his voyage around Africa in 500BC. The Vikings came to the Americas in about 1000 BC. Saint Brendan of Clonfert, also called the Navigator sailed in the 500s AD and wrote a book about it. God only knows where the Phoenicians travelled to.
There is another voyage recorded that began in the 1170s leaving from Wales and "supposedly" reaching the US, Madoc ab Owain Gwyndd made the trip with a crew and passengers. He supposedly arrived in the Panhandle area of Florida about 1173 and the people who came with him travelled inland and a group of "indians," the Mandans, were found during the Lewis and Clark era and their language had a lot of Welsh cognates. That is that they did not necessarily speak any pure Indian language. There are people in the S. Appalachian Mountains called the MELUNGEONS that no one knows where they came from or how they got there. They are not Indian, they are not Black, they are not a mixture of Indian, black or indian white, but they were there when the Cherokees got into the mountains.
In another little book I received from the author many years ago, CALALUS, he records the discovery of artifacts that date to the 70AD era, or early Christian times, of a Jewish/Roman group that got as far as New MExico and Arizona.
Barry Fell, in AMERICA BC, records many sites that indicate a really old, far older than the Renaissance, contacts with N. America.
There were Christian monks in Iceland when the vikings got there.
It would be difficult to figure out whether all of these trips were planned or if those who were in boats just got blown off course. The problem of entering the atlantic from the Mediterranean is that once you get a bit out into the sea, you get picked up by prevailing winds and currents, the same winds and currents that Columbus used in his initial voyages.
So, as I said, your premise is wrong. Check me out.