Obviously, people knew about the legend of Lief Ericson and his mysterious "Vinland" at the time of Christopher Columbus, as it is mentioned on the Piri Reis map of 1516.

They did not. The medieval saga literature of Iceland was by and large not transmitted outside Iceland nor translated until the 17th century. Vinland is not mentioned on the Piri Reis map. Perhaps you're thinking of the "Vinland Map" which is generally considered to be a 20th century forgery. Already the fact that it mentions Vinland alone is suspect, as A) Not a term at all for the entire continent, just one unspecific area, and B) Is only one of three named places in North America beyond Greenland (which is technically in North America) that is mentioned of in the two saga accounts, together with Helluland and Markland. Which today are generally interpreted as referring to Baffin Island and the Labrador coast. In terms of written sources, Markland is in fact the location in mainland North America that's most likely to have been visited, as a ship of Greenlanders returning from there went off course in 1347 and landed in Iceland, and that fact was recorded in several Icelandic annals at the time.

Which is quite another thing from the Saga of the Greenlanders and the Saga of Erik the Red, collectively the 'Vinland Sagas' (there is no 'Saga of Ericson'; both these mention him), which were written centuries after the journey(s) they purport to describe, which would've taken place around 1000 but are written down 1220-1250 or so, contradict each other on basic facts (like, how many voyages to Vinland occurred: One or five?) and have many fantastical elements and elements in common with other medieval folklore about "lands at the edge of the known wold".

The only account of Vinland known to medieval Scandinavians, and indeed an account which predates the Icelandic sagas in question by 150 years or so, is that of Adam of Bremen (~1070s), who mentions Vinland as an island in the far Atlantic the Danish king had told him of where grapes grow, hence the name. ('wine-land') And in this respect it does indeed not differ much from Thule, Brasil and many other mythical Atlantic islands.

Once Scandinavians started settling in North America, they'd discovered the Icelandic sagas and made sure to let people know they were there first. (e.g. Benjamin Franklin's letters mentions a Swede telling him about this) But the focus on Vinland really got started in the 1837 when the Danish historian Carl Christian Rafn published Antiqvitates Americana where he identified New England with Vinland, identified 'runic' inscriptions in Massachusets, the ruins of an early modern [windmill in Rhode Island](https://en.wikipedia.org/wiki/Newport_Tower_(Rhode_Island)) as a Norse fort, and other things. In America, this would synergize with impulses from the contemporary Viking Romanticism, to anti-Catholic sentiment and result in things like the Kensington Runestone hoax.

Source criticism did come into play but only around the 1910s. First the explorer Fridtjof Nansen published In Northern Mists. Arctic Exploration in Early Times in 1911, which compared many of these myths and took a very skeptical attitude to the Vinland Sagas. Then in 1912 an even more generally influential book, Lauritz Weibull's Kritiska undersökningar i Nordens historia omkring år 1000 ('Critical studies of Nordic history around the year 1000'). Although earlier historians in Scandinavia had always been forced to recon with the more obvious contradictions and fantasy elements of the saga literature, there had always been a kind of tacit assumption that there was always a 'kernel of truth' in there, some historical basis. Weibull (and his brother) took this idea to task; arguing (with examples) for how some of these stories really were probably made up out of whole cloth (except to the extent they built on earlier sources), but more importantly, that even if a historical basis existed, it was usually not possible to determine what it was in a source written far later. Emphasis was put on the contemporaneity of the source.

Although certain historians put more stock in the sagas (in this context, Mats G Larsson is noteworthy), by-and-large this is the dominant view. One would find very few serious historians in the past 50 years who'd for instance view the kings of Ynglinga Saga as historical, while as late as the 1930s-40s you can still find papers by leading historians assuming as much. (the saga-critical view did not come overnight; it took a generation shift)

Likewise, most scholars don't view the 'Vinland Sagas' as very reliable or useful as historical sources on Norse exploration of North America. There is evidence of contact with native Americans (whom they call Skrælingjar); they are for instance described correctly as lacking iron. However, they also called the Dorset (and later Thule) people of Greenland by that name, and all such details are equally true of them. And contact with Greenland's Skrælingjar is attested farther back as it's mentioned in the 12th century Historia Norvegiæ, recorded earlier than the 13th century Vinland Sagas. (and this is thus a case where the 200 years between those sagas taking place and being written down, matters a great deal)

That said, the idea of Norse people reaching North America was never discounted entirely. We still have Adam of Bremen's account, we have a credible and contemporary evidence of a visit to Markland (which at the very least had to have been west of Greenland, but which also fits with the saga description of being between Greenland and Vinland)

But what really turned this from conjecture into fact was simply that the remains of the Norse encampment ('settlement' is not really a good term as there's no evidence it was intended as permanent) at L'Anse-aux-Meadows in Newfoundland in archaeological excavations in the 1960s. So yes, this was the pivotal archaeological discovery.

But perhaps as a disappointment for the 19th century Romantics and Scandinavian emigrants, it did also mean that Vinland was not at all as far-flung as the present day United States. (In fact, L'Anse-aux-Meadows is neatly just about as far from Greenland as Greenland from Iceland and Iceland from Norway) The farthest any serious scholar today seems to venture would be the aforementioned Larsson, who believes Vinland was in Nova Scotia. Others think it's around Gulf of Saint Lawrence area, and others still go with north Newfoundland itself or the adjacent mainland.

More recently, indisputably European artifacts have been found in Baffin Island and at Dorset settlements. But also a carving of a medieval priest figure. While on the European side, we know for instance that some Inuit umiak canoes taken as trophies in Greenland in the 14th century by the Norwegian king, were still on public display in Oslo cathedral when Columbus went off on his journey. Some Scandinavians were aware of the strange people of Greenland from the 1100s through to the 1500s and onwards. But my point is, it's not really very relevant anyway; the Age of Discovery did not come about because Europeans suddenly realized there were new places to discover. Europeans hadn't explored the west coast of Africa before the 1400s either, and it's not for lack of knowledge the continent was there. There's not much reason to think things would've played out very differently even if the Vinland Sagas were widely known in Columbus' day.