There is a misconception in your question, when you state that “the Celts live only in the far north of the country and constitute just a small fraction of the Scots.” That is incorrect because there are not, and never were, any Celts in Scotland. The Celts were a collection of tribal peoples who flourished in central and western Europe from about 800 BC until 100–200 AD, before being subsumed by the Romans. Moreover, not only have the Celts been dead and gone for almost two thousand years, there is no evidence that any of them got within hundreds of miles of Scotland. For a bit more about this, you can take a look at an answer I posted here: Éamon O'Kelly's answer to If indeed people of Britain and Ireland are not that Celtic as it is wrongly believed, what are their origins, who were the indigenous people of these 2 islands?; or you could see Quora User’s comment to Paul Nicholson’s answer to this question, where she summarizes the state of play brilliantly and concisely.
But to address your core question: Why is Scotland considered a Celtic nation? (The same question could apply to Ireland and Wales.) The belief that the Irish, Scots and Welsh are “Celtic” was based on a series of misunderstandings, and now has been largely debunked.
At the turn of the 18th century, the Welsh linguist Edward Lhuyd discovered that Irish, Scottish Gaelic, Manx, Welsh, Cornish, and Breton comprised a single family of related languages. He theorized that the languages had originated in Iberia or in what is now France. Because the Romans had referred to some inhabitants of those areas as Celts, Lhuyd named the language family “Celtic.” Lhuyd’s interest was primarily linguistic, and he never suggested that the ancient inhabitants of the islands themselves were Celts.
In the 19th century, cultural nationalists in Great Britain and Ireland, influenced by the theories of Johann Herder, argued that if the ancient Irish, Welsh and Scots were Celtic speakers, then they must actually have been Celts, and therefore their descendants were the remnants of a Celtic “race” or Volk. That faulty logic led to the invention of the myth of the island Celts, a race of mystical warrior-poets.
In the mid-20th century, scholars came to agree that the Iron Age Celtic culture in continental Europe originated in the foothills of the Alps, and associated it with the great archaeological sites at Hallstat and La Tène. Because the sophisticated Hallstat/La Tène technology had spread across much of Europe, scholars assumed that the technology had accompanied migrations outward by the Celts themselves along with, of course, their language. Thus, if the peoples of the British Isles had been Celtic speakers, then of course they must have been Hallstat/La Tène Celts.
More recent advances in archaeology, linguistics, and ancient DNA research have killed that theory stone dead. There is no evidence (DNA or archaeological) that any Celts ever set foot in Scotland (or Ireland). It is now agreed that the Iron Age Celtic culture and the “Celtic” languages emerged at different ends of the continent (respectively, Central Europe and Iberia). The peoples of the British Isles are primarily descended from Bronze Age pastoralists (the Yamnaya) who originated on the Eurasian Steppe and who arrived in our islands 4,000 to 4,400 years ago. Their descendants picked up an early version of Edward Lhuyd’s Insular Celtic languages perhaps 2,300 years ago.
But the ancestors of the Scots (and the rest of us) were not Celts. The belief that the Scots, Irish and Welsh are “Celtic nations” rests on mistaken assumptions made by cultural nationalists 150 years ago.