You're quite right - the line about the SNP potentially not being in power needs to be hammered home more strongly. There will be elections in 2016 to the Scottish Parliament and that's when the Scottish Government proposes negotiations aim to finish and actual independence happens.
So yes, on day one we could see the SNP swept out of power, and that's quite feasible given that their support consists of a lot of folk who would traditionally support other parties but for the independence issue.
Obviously, though, the SNP would want to stand and win in that independent Scotland so they are probably unwilling to encourage people to think about the prospect of their removal from office. Meanwhile the Yes campaign in other quarters does talk about the scenario, but perhaps it doesn't get sufficient coverage.
It raises the curious paradox of an argument that says if you don't like the SNP, then independence is the way to get rid of them. It's neatly summed up by this taken from <a HREF="http://www.indyposterboy.info/" rel="nofollow">an unofficial Yes poster website</A>:
<a HREF="http://www.indyposterboy.info/scottish-independence-referendum.asp" rel="nofollow"><img src="http://www.simonvarwell.co.uk/wp-content/uploads/2013/07/a4_hate_snp-212x300.jpg"></A>
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