What you are proposing is essentially using a different number basis, and the simplest applicable one would be a basis which expresses numbers using whole numbers{...,-1,0,1,2...} times pi. This allows you to exactly express pi as just 1, and all such using only whole numbers which is nice. It also unfortunately means that simple things like y= 1_{decimal} x would have to be expressed using the relative to basis transcendental number pi i.e with precision problems. And every basis you could chose behave like this.
Different basis have different advantages, same as different function basis. The classic example would be that in a standard basis, its easy to add and subtract, but more costly to multiply, divide or factorize, while in the prime basis the former is expensive as hell, but the latter is trivial.
As a result, rationals and pi in a sense disjunct domains. You cannot express either using less less than an infinite number of the other, and the same holds for combinations of a rational and pi. Numbers which behave this way relative to each other are more common than the rationals, and pi is just the most common example.
It does lead to a rather neat requirement for the fundamental physical constants though.
The reasoning goes like this, imagine that a model K2 of physics could be described using two constants, a, b. gravity and the speed of light say. Now lets say we managed to prove that a = 2b and therefore that everything predicted by model K2 can also be predicted by model K1, which just uses the coefficient b. K2 is equivalent to K1, sure, but only one constant would then be fundamentally required, and if K2 is sufficient to describe all of physics, physics would only have one fundamental constant.
The same reasoning would hold if a=b^2, and so on. But, if the function required to express a as a function of b requires infinite information, this does not meaningfully apply, as this will always apply to every pair of numbers. Meaning that we know that if the fundamental constants of a model of physics does not lie in disjunct domains in the sense above, there is a simpler version which has fewer constants. For example, since pi has infinite information, if a=pi b, then the simplification cannot be meaningfully made without introducing pi as a fundamental. More generally this also fundamentally means that true physics cannot be expressed using finite precision if ideal grand unified theory as more than one fundamental constant.