The die has no memory of what has been rolled recently, or how many times it has been rolled since the last 1. The average will always be six rolls starting "now", no matter when "now" is.
Something's going wrong here with my mental image.
I get that at any particular moment, the probability of the next hash being valid is (essentially) the same as the previous hash.
Yet after the fact, they do appear every 10 minutes on average.
If I was a bad dart player throwing darts at a very long timeline where there were regular marks every 10 inches, I'd expect half of the darts to end up closer to the right mark and half the darts to end up closer to the left mark.
In this case the marks aren't evenly spaced, but I have trouble seeing why that matters over a long timespan since the intervals average out to once per 10 minutes.