Is it because pi isn’t measured, but calculated? The wikipedia article (https://en.m.wikipedia.org/wiki/Physical_constant) makes a distinction between a mathematical constant and a physical constant, stating that the latter cannot be calculated but instead needed to be measured experimentally… Pi could be measured experimentally, but it has an exact definition and can be calculated outside of any experiment.

If you run Buffon's needle a lot you come to the conclusion that we live in a locally Euclidean space. That's fine and good. We also live in what you might consider a "locally Newtonian" world, but when things get very big or very small, the Newtonian approximation breaks down.

The ratio of a circle's circumference to its diameter (or equivalently the sum of triangle angles) has the same problem. If you want general relativity to work, then we need to live in a curved spacetime. Depending on whether that spacetime is positively or negatively curved, the angles of a very large triangle may add up to more than or less than pi radians.

Buffon’s needle assumes a flat space, while a non-Euclidean space or geometry would affect the probability leading to a different value other than pi. You can treat the space around us as Euclidean, but that isn’t true for every part of the universe.

Up until 2019, it sort of was, via the magnetic constant mu_0 = 4 pi x 10^-7 H/m exactly. Unfortunately the 2019 revision of SI redefined it.