Good question. It is generally assumed it does, although it does sound a bit odd when you think about it. Apart from the obvious that 'good' and 'big' have some similarity in meaning, I can't find much of an explanation.
The only thing I could find in Mark (2003) is that mòr can sometimes mean 'long' when referring to time, as in bliadhnaichean mòra, but that's not really the same thing.
The English word great does show that one word can mean both 'big' and 'excellent'.
I did find a couple of cognates that could also mean something like 'good'.
- Proto-Germanic *mēraz is defined as 'great, excellent; famous', although it is related tow words that clearly mean 'big'. The entry for the related word *mērijaz specifically list the Old Irish mór as a cognate.
- edil, the dictionary of Old Irish lists 'great' as the first meaning of mór but I suspect this is 'great' in the sense of 'big'. The first section covers 'big, great' and the second 'mighty famous'.
GPC, the big Welsh dictionary gives a range of meanings for mawr fairly similar to the other languages, 'big', but also 'long' of time and (meaning f) 'great, important, significant, powerful, influential; grand, noble, remarkable, renowned (often corresponding to ‘the Great’ of kings, emperors, &c.), self-important, proud; active, enthusiastic, skilful, adroit, knowledgeable; favourite; striking, strange, surprising, wonderful, awful, terrible'. But nothing that really fits here.
So I don't really think we can do any better than 'if great can mean both 'big' and 'excellent', then why can't mòr?'
This looks like an interesting distinction between what people actually do and what people think is correct. Not knowing the exact phrases, I looked in Wiktionary. Italian Wiktionary (presumably written by people who speak Italian) says cheers means alla salute, and French Wiktionary (presumably written by people who speak French) says it means à votre santé. I guess these people wrote that and then said salute or santé next time they were in a bar.