I've found that people who promote Blair and even Cameron as Scottish without mentioning that actually they think themselves English tend to be nursing a grievance against Scotland for whatever reason.
Otherwise we have Brown who was unequivocally Scottish and was often attacked on that basis during his premiership ("One-eyed Scottish Idiot" etc).
Otherwise in the 20th Century we have James Ramsay MacDonald (1930s for 6 years), Lord Douglas-Home (a year and a half in the 1960s), Henry Campbell-Bannerman (2.5 years), and Arthur James Balfour (3.5 years).
So 13 or 14 years of the 20th century, Britain suffered under Scottish prime ministers, only slightly above what we might expect given Scotland's share of the population of the UK - this could be explained by Scotland then-increased share of MPs.
It doesn't seem that Scotland dominates UK politics more than any other nation. However, it is certainly noticed and commented upon or even resented when Scottish people rise to high office, perhaps that's why it may seem so to some.
In the current political climate with the domination of the SNP, it seems like the possibility of Scottish MPs attaining any kind of high office, membership of the cabinet, or part of the governance of the UK has entirely vanished for generations with who-knows-what consequences for the union in the future.
Edit: The self destruction of Labour in Scotland has pretty much guaranteed this.
"Actually, Arnold Schwarzenegger isn't Scottish" - "HA! No True Scotsman!"
I'm sure you can see the problem with using this "fallacy" to claim that anybody is Scottish even if they don't self-identify as Scottish?
Your position appears to be "they may identify as English, say they are English, and sound English, but because it suits my agenda of suggesting that the UK government is run by pernicious Scots, they are actually Scottish" - which is ridiculous.
The fact that Blair and Cameron do not identify as Scottish is sufficient reason to say that they aren't Scottish, and does not invoke the fallacy.
It's possible in the future one or more self identifies as Scottish. Alternatively, let's say Cecil Rhodes had said, I'm not English, I'm South African, would that have made him South African and not English?
(Stewart Lee).
On the other hand, he was, as you admit, born in Scotland, spent many years of his childhood there, including (I may be wrong) the formative secondary school years (ages 13-18), and I don't think it would be crossing the line if some proud Scot were to claim him as one of their own.
Your second line vaguely defines a 'people' but given the context, I am assuming you suspect me as one of them, so for the record, I am English-born of a largely Scottish origin family, foolishly proud of my Scottish heritage, and do not nurse any grievance - quite the contrary. Indeed, I voted for both recent 'Scottish' prime ministers. I've never noticed a grievance in others claiming Blair as Scottish but will look out for it.
Your last sentence makes a sad but good point.
My second point was not aimed at you in particular, apologies if it came off that way, just that it is common for some (perhaps right wing) people to tediously repeat the "Blair is Scottish, Scots have an outsized influence on the UK" mantra without reflecting that actually it isn't as clear-cut as they state and that Scots don't have an "outsized" influence on the UK politically and never have.
They have an influence in proportion to their share of the population, over the historical piece - sometimes higher and sometimes lower (as now) but it all comes out in the wash.