I don't have a source to support the OP, but I would think there is a significant allowance for course correction that reduces the required precision. It's not a purely ballistic trajectory.
The mid-course correction is not because you used an imprecise value of pi. If that's all it were about, you'd just use a precise pi from the beginning and then not even need the mid-course correction. Missions have failed because the rocket engines failed to start up for the mid-course correction. Unnecessarily risking that because for some reason you insist on using far fewer digits of pi than your computer is capable of would be insanity.
The context of my statement/question was definitely "given the precision that can be achieved with available precision" and not with an arbitrary imprecise constant in the calculations. As you say, the mechanical aspect of the launch and transfer orbit insertion is going to be several orders of magnitude more imprecise than the mathematical goal - that is what the mid-course corrections are for (aka the difference between theory and reality)
The comment I was originally responding to said "Has anybody needed more [digits in pi] than 3.14?" and "NASA's rocket trajectories are less precise than [one part in a thousand]", both of which I take issue with. You then stepped in to defend those statements against my objections. I'm glad it turns out you don't actually agree with those statements, but hopefully you can see my confusion.
