When considering 3 first numbers in pi, we get 'fixed numbers' 314. However, three integers can have 999 =10^3-1 versions, of which only one 'belongs' to pi. Thus, for n first digits of pi we have 10^n-2 versions of integers which do not belong to pi ratio being 1/(10^n-2)-> 0.0, when n increases. Thus, for increasing number of n, the probability of finding n random numbers in n 'fixed' numbers of pi approaches zero?
There is no reason to expect randomness in digits of $\pi$
It is a common mistake that an infinite sequence of numbers without being eventually periodic must include all combinations.
For example $$1, 1,2,1,2,3,1,2,3,4,1,2,3,4,5,...$$ which is not eventually periodic and does not contain $55555$