My guess is that the Galwegians moved to Glasgow in large numbers at some point in history, bringing their dialect with them, and eventually Glasgow Galwegians comprised the majority of Glasgow residents. Then the name became associated only with the city when Galloway no longer existed as a political region. Perhaps there is a Scotsman out there who could confirm or refute that?
I am a Glaswegian. This etymology is interesting but extremely hard to believe. Historically, the demographic shift went the other way, i.e. from the Borders to Glasgow (and indeed, from everywhere to Glasgow, which was where there was work from the industrial revolution onwards, as well as educational opportunities etc).
That said, one of the dialect words for Glasgow is 'Glesgay', the last syllable of which rhymes with 'way', although more common now is 'Glesca'.
It seems to me many towns have names for their citizens which defy logical analysis. The inhabitants of Edinburgh are 'Edinburgoynians', not 'Edinburgers' which would seem more natural. The inhabitants of Bristol are, I belive, 'Bristolians', whereas 'Bristolites' seems more natural.
Who can say ? Perhaps these terms were invented by some bored academic locked up in a cloister for too long.
Folk in Moscow are, I belive, 'Muscovites', bythe way.