I have to admit - this is odd. Because ALL my restrictions are on (as in, Restrictions is turned on and all the buttons show green). If I don't allow Safari, the whole icon disappears, so I can't even get into it.
Marilyn
Actually a green button shows that the restriction is not on. It's allowed, as the section header says. {I'm sure that's what you really meant}
I can't see any setting in restrictions that 'says' it wold block private browsing; but here are my two guesses.
If Websites under Allowed Content were change to Limit Adult Content, it might block private browsing as well, under the assumption that if an adult does not want a child browsing adult content, they also don't want the child being able to hide what they do browse.
Accounts under Allowed Changes. If account changes are locked, maybe Safari's modes get caught up in that. Seems unlike, but. . .
Anyway, I have not tested any of these. Feel free to turn ON/OFF various restrictions and see what they do.
BTW, I have restriction enabled, but have not actually applied any restrictions. Yes, that is possible. I do it so no one can set the restrictions code to something I don't know, and lock me out of my own iPad's settings.
And private browsing works for me, though I rarely use it.