Superscripting the lower c in MᶜCartney is a bad idea.
It makes the letter too small and hard to read. It makes the letter size half the size it should be.
In printed text, rather than use superscript, it is easy to push up the c and keep it the same size by increasing the leading. That means just adding some space under a normal c. It is very easy to do and no real cost.
Superscripting the c on the web is also easy using css (stylesheet). Again, no real cost is involved.
I believe they do not want to do this because
a) It looks bad and makes the name hard to read for visually impaired
b) It is a bit fiddly to do
And Karl MᶜCartney is a wazzock because if he really wants this, he should start with his own website. It is riddled with a normal lower c throughout.
http://www.karlmccartney.co.uk