> when more reasonable modern default choices for all purposes outside of high performance applications would either be a rational composed of bigints, or a bigdecimal, or maybe a generating function for the coefficients of a Taylor series expansion or something like that if you really want to get nuts.

Rationals of bigints doesn't work in general. First of all, you still have to do approximations if you take e.g. a square root or a sine. And even if you have an algorithm that is closed under rational numbers (like the Simplex method for solving linear programs), the number of digits you need in your integers blows up unless you have a tiny problem.

My rough sketch is that inv(A) involves det(A) in the denominator, and the number of digits in det(A) is exponential in nature (n! if A is n-by-n).