Was it more accepted than Polynesian or Basque theories, or was it totally fringe?
Just a week ago I answered nearly the same question; At what point in history did the Europeans realise that Vinland and Canada were in the same place
However, transatlantic contact was a fact. There was never any doubt Scandinavians had been to Greenland, which is in North America and had contact there with North American natives.
"Vinland" had been amplified into a big thing in the 19th century who wanted to have Leif Erikson as an ersatz Columbus, a North European discoverer of the American continent. Preferably even the present-day United States. But the sources describe Vinland as an island or area, not a great land mass. (and in fact out of the places mentioned, "Markland" is the one most certain to have been on the mainland, in Labrador, possibly around the Hamilton Inlet).
But this viking-romanticist and nationalist ideas about Scandinavians discovering America had great appeal in the 19th and early 20th century. see e.g. the the Leif Erikson monument in New York, created in 1939. It's clearly not very historically accurate; it has Leif wearing a winged helmet, in the best Wagner-esque style and dramatic pose, a giant viking ship and rune-inspired lettering. Leif is credited as discovering "America". The reverse is a copy of the Tune stone, which actually predates the Viking Age by a whopping 500 years or so. They didn't know that Leif had actually been to North America beyond Greenland at that point, they certainly did know that vikings did not have winged helmets and that the Tune stone was 500 years older than the Viking Age. It's about accurately portraying history though, it's about the romantic, nationalistic myth.
As mentioned in my other post, the sagas, including the Vinland Sagas, had begun to be treated much more critically by academics in the 20th century while the World Wars killed off the romanticized viking imagery in popular culture quite a bit, particularly in Europe. The Nazi appropriation of such imagery was just the last nail in a coffin that'd started to close in the 1910s. But it's hard to completely kill a popular thing, and it's telling for instance that some neo-Nazis and other groups continue to use "Vinland" as if it were a name for the entire continent. (or any specific known place) But romanticized views remained more popular Scandinavian immigrants in the United States.
So by 1960 when investigations started at L'Anse, the public wasn't quite as viking-crazy as they'd been and Scandinavianist historians were more inclined than ever to be skeptical of the sagas account. Which is not to say most outright denied the possibility, but that what parts of the sagas (if any) had any truth to them was not possible to ascertain. That said, travel to northern Newfoundland, where L'Anse-aux-Meadows is, always had a better plausibility to it as it's about 1200 km from the southern tip of Greenland, the same distance as Greenland-Iceland and Iceland-Norway. The East Greenland Current also makes it quite plausible that a sailor headed towards Greenland who was too far south, could easily end up in Labrador or Newfoundland. (which has some support in the Sagas) The problem with that tough, is that wild grapes don't grow in those areas, invalidating the Saga (and Adam of Bremen) account of the name.
I don't know which Polynesian or Basque theories you're referring to (it's not how you'd compare a prehistoric migration to a historic voyage anyway) but the idea that Vikings had made it to North America was not so fringe. They had after all done so, with Greenland at least. It was both plausible, and there was a written account. It was just not a reliable written account. Many ideas of what and where Vinland was were far-fetched. It remains very unlikely that they'd have made it as far as New England and other places the 19th century enthusiasts wanted to place Vinland. Even knowing about L'Anse-aux-Meadows now, mainstream historians don't speculate that they'd have gone much further south than New Brunswick or Nova Scotia. (and as said in the other thread, those who point to such locations tend to be the mainstream historians who take the sagas at face value more than most)
As /u/y_sengaku mentions in answer in the thread above, the other places mentioned like Helluland and Markland were perhaps more interesting to later Scandinavians than Vinland was, prior to the 19th century 'decision' that Vinland meant North America.
An interesting bit of textual evidence is a short entry in the Icelandic Skálholtsannáll from 1347, where a ship is mentioned having drifted to Iceland when going off course from Markland to Greenland. To many readers, the offhand way in which this is mentioned seems to imply that Greenlandic travel to Markland - possibly to get wood - had perhaps been commonplace.
Today we also have finds (e.g. a figurine and crucible) from several sites on Baffin Island hinting at Scandinavian presence on what was probably Helluland - land of cliffs/flat rocks.
So there's more to it than just Vinland.
It would be useful to distinguish the reception of the thesis between the academic literature and the more popular and local history.
1. The Popularity of the trans-Atlantic Norse voyage thesis among the researchers
It roughly also concern with how we evaluate the written 'evidence' of the voyage, two Vinland Saga texts that come from the 13th century Iceland, allegedly written in the parchments after more than 200 years after the original event. As summarized conveniently by /u/Platypuskeeper in At what point in history did the Europeans realise that Vinland and Canada were in the same place, the first half of the 20th century was, so to speak, 'the winter period' of the utilization of Old Norse-Icelandic sagas as historical sources. Put it simply, the researchers of this generation, such as Swedish Weibull brothers, Norwegian Halfdan Koht, and Icelandic Sigurður Nordal regarded sagas rather as learned literature and apply the source criticism to them, focusing on the trace of more contemporary elements found in the texts, in contrast to their predecessors who had put emphasis on their historicity on ground of the oral tradition.
Gísli Sigurðsson compiles the table of 12 saga scholars on the identification of various place names of Vinland saga texts, including Vinland itself, but there are more than 30 years' gap between the last research (Hermanson 1936) prior the discovery of L'Anse aux Meadows and Ingstad's report in 1969 (Gísli Sigurðsson in Fitzhugh & Ward 2000: 233). It suggests that there was no notable study published on this topic between the middle of the 1930s and the discovery in 1960s, at least in his understanding.
2. the Reception of Vinland in Popular History in the First Half of the 20th century
It is worth noting, however, that the 'historical' discourses in the popular history was totally different matter, and the rise of such a more source critical approaches to the sagas, as I summarize above, seemed not to affect their discussion so much. Inga Dóra Björnsdóttir argues that the popularity of the Leif Eriksson as a 'real' discoverer of America over Christopher Columbus did not decreased in the first half of the 20th century especially among the Americans of Scandinavian descent, in light of the Eugenics. Some Eugenics thinkers categorized the Scandinavians together with the Anglo-Saxons as the 'Teutons' over other categories of the Europeans, thus Leif 'of the Teutons' was more suitable 'founding father' for them than Columbus (Inga Dóra Björnsdóttir in Wawn & Þórunn Sigurðardóttir 2001: 220-223). Their discourse, however, did not attract much attention from common non-Scandinavian Americans.
Not only in USA, but also in Iceland, further in Canada, Leif was a popular historical figure in the first half of the 20th century. The latter two countries were bridged by Icelandic immigrants in some areas in Canada like Nýja Iceland (New Iceland) since the late 19th century. Some nationalistic Icelandic poets, such as Sigurður Jón Jóhannesson (1850-1933) kept reciting the voyage of Leif in their poems, and some of these poems are in turn not only recited in their homeland, Iceland, but also in Canada, especially around Winnipeg where some Icelandic immigrants settled. Kirsten Wolf analyzes such a popularity of Leif among some Canadians of Icelandic descent primarily from a point of view of the identity crisis of the fragmented Icelandic immigrant community in Canada during that period (Wolf in Wawn & Þórunn Sigurðardóttir 2001: 207-19). Regardless of the trend of the new historical research on sagas in Scandinavian countries, the immigrants of Icelandic-Scandinavian descent carried on commemorating Leif as a symbol of their immigration.
References:
