The question seems to presume they were heading somewhere, as if the Americas were a goal to be reached.

But early humans weren't heading anywhere, their spread was just diffusion. When a camp had too many people for the local area to feed them, some would move off a few miles and set up another camp. They aren't going anywhere in particular, just moving away from a densely populated area.

Around the time humans moved to the Americas, there was a huge landmass between Russia and Alaska, called Berengia. This area was colonized from east Asia and supported human populations for over 10,000 years before sea levels rose and it vanished under the sea.

The interior of Alaska was frozen over and not very inviting for human settlement, but as population pressure grew in Berengia, some people migrated south along the coast across Canada and eventually Washington and Oregon. The large mass of water of the Pacific mostly kept the coast clear of ice, and there were rich food resources in the form of marine mammals and fish.

These people had no maps and weren't aware they were exiting Asia and entering North America, they were simply doing what their ancestors had done for a million years - even before our species - which was to move camp when the previous one became too crowded.

Regarding the maritime route you mention, there's a huge ocean between Africa and South America. We have absolutely no evidence that humans even had boats at the time humans first entered the Americas. Even if they did, it could hardly be ocean-going boats capable of surviving the long trip and storing food and water for months. We are talking about a time that even predates pottery.

Then there's the question of why. They didn't know South America existed, why would someone build a boat, store months of food and water, collect wife and children, and head out into nowhere? All you can see standing on the coast of Africa is empty ocean.

There was a theory (never taken very seriously) that Europeans may have crossed the Atlantic to North America during the Solutrean, 20,000 years ago. Again, while evidence of any kinds of boats is completely missing, the reason why it wasn't rejected outright is because there is land between the two continents at those latitudes. A series of islands chained across - England, Ireland, Iceland, Greenland, Newfoundland. During the ice age, with lower sea levels and more land exposed, with huge ice sheets you could walk over, with much smaller watery gaps along the path -- it was barely conceivable that humans could have worked their way across the ocean. Not that there's any evidence they did.

But these features are missing further south between Africa and South America. So it's really not possible for humans to have crossed that ocean in that time frame.