As another follow-up: Cabot himself was ethnically Italian/Venetian (despite sailing for England), so doesn't that argue against OP's idea?
Along the same lines, wasn't Amerigo Vespucci the first major explorer to land on the North American continent?
With people already addressing the fact that John Cabot was himself in fact Italian and as such theory appears contradictory, i tried looking up the other part of the statement, that it was popularly believed that Cabot was discoverer of America.
I am not sure if my approach was correct but i decided to look up various 18th and 19th century encyclopedias and check what do they say. The encyclopedias of those times are full of inaccuracies, but it probably best shows the then consensus on who the discoverer was, which is what we are trying to find out.
First I looked up Diderot's Encyclopedia, written in 1751, for entry America, and it names Christopher Columbus as the discoverer. The English translated article here, and the original scan here.
Encyclopaedia Britannica, 4th edition (1810) in America section on page 34 also identifies Columbus as the first to arrive
The same edition, different volume on page 39 entry for Cabot does not even mention John Cabot at all, but his son Sebastian as "discoverer of continent of America". I looked up Columbus in the same edition (pages 170-173) and it credits him as "first discoverer of islands of america"
As far as i gather Columbus was always considered the first "discoverer", the only difference then, and now, being considering who actually discovered the North American continent first. Columbus remained at the Caribbean islands, while John Cabot in 1497 discovered Newfoundland and soon the mainland. For /u/nighthawk_md question about Vespucci, he came to America first in 1499, after Cabot's expedition as the letter claiming his voyage in 1497 now seems a forgery and possibly never happened. Even if true that one and his definite 1499-1500 and 1502 voyages were only around southern America anyway, so not in contest to finding North America
So to sum up, I don't think there was ever any doubt Columbus was the "first". There might be some confusion if someone claimed Cabot was the first to discover North American continent, which is true, but I do not think it was ever so serious to cast doubt or anonymity on Columbus
Meh, although Columbus Day was appropriated by the Italian American community as their celebration of heritage, even not knowing anything at all about the theory mentioned I would discredit the hypothesis: John Cabot was Italian, just like Columbus. John Cabot is an anglicization of "Giovanni Caboto," who was a citizen of the Italian Republic of Venice, although his actual ethnicity and origin in Italy is disputed (records seem to indicate he was born in Gaeta, in central Italy).
As a followup question, if the claim OP listed was true, why was the New World called Columbia and not Cabotia?