Nit to be pedantic... but do you mean they speak with a sort of Scottish accent, or do you mean they are actually using words from Scots, Doric or some other Scottish dialect?
But technically it's just Scottish the language as it is referred to in English there's no need to add Gaelic at the end. Gaelic would be "Scottish language" in Scottish.
Just as Irish is a language and in the Irish language Irish is Gaeilge.
It's pretty confusing but I think I got it right! Koralatov could explain better below. I'm about 4th generation Irish folks left problem late 1800s from Co. Monaghan.
Slán!
GordonS is right: Gaidhlig is only ever referred to as "Scottish Gaelic" or, more commonly, just "Gaelic" by Scots on a day-to-day basis. The concept of "Scottish" as a language is pretty murky and is generally not used seriously. Even in these days of awakened Scottish national identity and independence referendums, we all still speak English of a Scottish variety and call it English.
Calling Gaeilge "Irish" and Irish English "English" makes a lot more sense, frankly.
Many modern english speakers derive from a Scots root rather than a purely English one.
E.g. Scots has a plural form of you, 'yous' and other differences.
Scotland has many other dialects like Doric and accents and other languages like the Gaelic of the Highlands.
What is considered a language or a dialect is largely a political construct. By the way you put it, it seems to be the case here too.
It's a stretch to argue that "many modern English speakers derive from Scots root". At this point in history, most Scots speak Scottish English with some borrowing from a Scots root -- not the other way around. This is evident in the fact that the vast majority of a modern Scottish person's vocabulary is from mainline English and intelligible by someone from London. That doesn't happen by Scots borrowing vocabulary from English -- it happens from English absorbing vocabulary from Scots.
This is the situation for the vast majority of Scottish people. Coming across those who don't fit into the English-with-Scots-influence only reinforces this. Doric is arguably a descendant of mainline Scots which has converged on English, rather than the other way around, and means it's sometimes difficult to parse even for other Scots. Quite a lot of the Borders also speak a dialect which derives from Scots which has converged on English, and is difficult for other Scots to parse. They sound very similar to each other to someone who speaks neither, but are actually quite distinct -- mistaking someone from Selkirk for an Aberdonian will cause offence!
The Wikipedia has a pretty good article on Modern Scots, though I think it overstates the pervasiveness of modern Scots: https://en.wikipedia.org/wiki/Modern_Scots It's also worth noting that most speakers of Modern Scots also speak Scottish English, and code-switching takes place frequently and fluidly.
Gaelic is spoken by ~57,500 people in Scotland. Some of them have English as a second language, having been raised with Gaelic as the language of the home. In some cases this does show their English usage. One common tell is slightly unusual phrasing -- one typical and quite common usage is "You will be having tea?" rather than the more common "Will you be having tea?" Even then, they're still speaking a variety Scottish English and not Scots.
tl;dr: Every Scottish person speaks at least one of the dialects of Scottish English. Some also speak a dialect of Modern Scots. An even smaller number speak Gaelic.
Sources: Highlander born and raised, living in Aberdeen for >10 years. One parent is a Gael whose second language is English, the other parent is from the Borders whose first language is that variety of Modern Scots. Native speaker of Scottish Standard English, with a little time spent studying linguistics and an interest in the subject.
You're bang on about Doric being difficult for people from elsewhere in Scotland; I have family from the central belt and they really struggle to understand some folk up here!
https://www.youtube.com/watch?v=0isMWmwfO34
and Aye Can's Doric page [2] is part an audio map of Scottish Dialect
[2] http://www.ayecan.com/listen_to_scots/north_east.html
Sheena's Granny's Doric [3] has a flavour of the musicality of Scots.
[3] http://media.scotslanguage.com/library/audio/ayecan/census_n...
The Scots sort of sang their words, especially the Gaelic speakers, Orcadians and Shetlandic.
https://www.youtube.com/watch?v=v37bgydws0E
The BBC's postwar campaign to establish Recieved Pronunciation made the UK largely comprehensible to each other but at great cost to it's unrecorded linguistic diversity - at the time sadly considered vulgar.
Lang may yer Lum Reek ! ( Long may your chimney's smoke - a Fifer greeting )
Dinnae fasch yersel. - Edinburgh's Don't worry yourself ( Keep Calm )
It would be interesting and I can certainly see how we Canadians and Scottish people are similar especially here on the east coast. But there are also big differences I'd say Scotland is more left of center politically than Canada usually is.
But after trying so long to be independent I can't see anyone in Scotland wanted to join Canada or any other nations. Maybe the EU eventually but not a union with another country.