From your addition of his being a hat-maker, I found this, which also starts to tie in with Hugh's suggestion - although there were probably hat makers in conurbations of any significant size-
"That a certain progress was being made is shewn by the growth of the population. According to a census which the magistrates ordered to be taken in 1763 the number of inhabitants was 28,300, an increase of 2754 in six years. At the same time, from the list of the city's exports in 1771, given by Gibson in his History, the list of goods actually of Glasgow origin is by no means lacking in variety. It includes ale, books, coal, cordage, glass, hats, linen handkerchiefs, wrought iron, tanned leather, sail-cloth, soap, candles, woollens, and herring. [Gibson, History, p. 226. Curiosities of Citizenship, p. 145.] Wrought iron—spades, hoes, axes, etc.—was produced by the Smithfield Company at the Nailree established in 1737 at the Broomielaw, and exported to the annual value of £23,000. In 1748 there was set up on the site of the present James Watt Street at the Broomielaw a factory for the making of glazed pottery, or Delft-ware, so named from the place of its origin in Holland. [The earliest Glasgow pottery, however, was made at the old "Pig-house" off Gallowgate.—Burgh Records, 8th May, 1722.]
(My break - AD)
Hat manufacture appears to have been fairly extensive. To Maryland 557 dozen were exported, and to Virginia 2971 dozen. [There was a hat factory near St. Andrew's Church. Burgh Records, 31st Aug., i 769. Perhaps the largest hat-making firm was that of Thomas Buchanan of Ardoch on Loch Lomondside, whose eldest son, John, was I.P. for Dunbartonshire from 1821 till 1826, and built Balloch Castle and Boturich Castle.—Curiosities of Citizenship, p. 184.] There were exported also, strangely enough, considerable quantities of snuff and manufactured tobacco. Similarly, among the exports to the West Indies—Jamaica, Barbadoes, Granada, Antigua, Tobago, and other islands, there invariably appears a considerable amount of refined sugar—a case, one might have thought, of sending coals to Newcastle.
As you can see, Glasgow was the main import/export centre in Scotland from/to North America, being on the west coast and having the relatively sheltereed Clyde estuary, the Firth of Clyde, protecting the navigable parts of the River Clyde at the Broomielaw etc.
(For similar reasons and others, it was the British port for the two "Queens" during World War 2.)
Of course, this is not proof that your ancestor came from that area; but keep in mind that in the later years of the 19th century, many Glaswegians followed the export routes of the ships and locomotives built around there.
On the LDS Family Search site, there are a few Barclay and variant Births around 1758, in North Ayrshire and Renfrewshire - but none are conclusive.
There might be confusion also in the actual calendar year, due to the uncertainties of which, Julian or Gregorian, might have been in local use at that time!
Regarding the placename, several years back, I helped another person who was trying to track down a location reputedly in Scotland, called "Tiles".
This was on the basis that an elderly relative in the USA could recollect an even older aunt, who was an early migrant there, saying that she came from "The Tiles" in a "broad Scotch accent".
The researcher managed to locate a "Tiles Farm" in North East Scotland; but it did not fit other information.
Eventually I managed to locate that the gggg?-aunt had been born in "The Dales" of Yorkshire; thus it was a "broad Yorkshire accent".
This also fitted in very well with the other known facts of the family.
Incidentally, I am not suggesting a similar error for your ancestor, just the extent that words can be apparently mis-pronounced, leading on to innocent errors in the written records.