There are plenty of series which do operate like this, but you still get performance gaps. Look at IndyCar, where Rodger Penske is king with the same cars.
There's also an argument to be made that part of a drivers job is development. Having good communication with engineers and feeding back the issues with the car, how the car is doing, and being able to give the engineer as much data to work with as possible. These are all part of a drivers skillset.
Having equal cars doesn't necessarily mean the best driver wins, even if you do equalise everything. Some drivers suit different cars better. Vettel likes a stable car that he can really really abuse. The slightly lazy power delivery gives him the confidence to deploy power that others do not. But when they switched to the turbos, with the spiky power output he struggled. Barrichello was a right foot braker, who struggled with the Ferraris which were setup to be better with the left foot (which suited Schumacher). And then there's the different tracks - Irvine always ran well at Suzuka, etc.
There's just too many variables to stick them all in the same car and assume the winner is the best.
