I think the SNP are making this point to an extent, and do often draw on Scandinavian or Nordic identity as a regional equivalent. Though I sense there is a slight tension in the party between a new wave of accepting this regional Britishness as a part of people’s prospective post-independence identity on the one hand, and on the other hand a strong dislike of anything “British” like the monarchy, BBC or empire that many in the SNP have long stood against.
This is why the multiplicity of pro-independence standpoints is a good thing – socialist, pro-business, Green and of course nationalist arguments all point to independence from different starting points, and all would embrace British identity (or not see it as a problem) more than many within the SNP.
The result is that some people – particularly neutrals – are not distinguishing, as you say, between Great Britain (the cultural and geographical entity) and the UK (the state); though many increasingly are as the message gets a little clearer over time. And I think it will continue to do so.