The question about other bases is insightful: because what appears more frequently in one base might not appear more frequently in another base; indeed, the pseudorandom nature of the digits appearing would implicate that unless there is a relationship between pi and your particular base, you might expect that for any string of digits there is a random frequency, AND a random distribution.
Which would then imply that you will see zero eventually overtake every other digit, and then go back to least frequent again.
But is that a good assumption?
If I wanted to look for a relationship between pi and base ten, I think I would look at the Julia set fractal, where PI appears. I would base a calculation of pi off that. Then I would look to see if the spiral that was involved in that calculation had any relationship to a frequency of ten:1. If I found no such relationship, then I would look for a DIFFERENT location, and a different calculation of PI.
IF somehow I ran out of all the calculations available in the julia set (I suspect the number of such calculation are going to be infinite, but with classification,suppose we could cnfirmably run out), that's okay. The Julia set is only a 2-D fractal, and one class of fractal. There are others.
-- (March 12, 2019) on Frequency of Each Digit of Pi