I think you're confusing dialect with language here. Scots and Scottish English are distinct from one another. The latter does borrow some from the former which leads to confusion. Scots
is a language, historically equivalent and closely related to English but still distinct.
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.