Indeed. It’s not the birthplace of the PM that’s the issue (indeed the current incumbent was born outside the U.K. altogether), but the location of the constituency they represent.
This is because in Scotland, many functions of government are handled at Holyrood, where MSPs are the representatives not MPs. Therefore in the event Swinson became PM, she’d have a say on Education, Health, law and order and so on in England but not in Scotland. This might, for example, lead to a PM privatising the NHS in England, while knowing that the hospital serving her own constituency remains in public hands. For example.
Devolved versus reserved powers:
What are the powers of the Scottish Parliament? - Visit & Learn : Scottish Parliament
This asymmetry to differing degrees also applies to the other devolved institutions, depending on which powers are devolved. So that’s Wales, Northern Ireland, and yes London, although the mayor and assembly are seldom mentioned in this context, but of course the same argument holds: an MP in London represents a constituency where certain (albeit limited) matters are devolved whereas an MP in Devon, or County Durham or the West Midlands does not.