A lot of the parameters that enter their equations are probably measurements, like the gravitational acceleration, properties of some material, and so on. The numerical solutions to their equations have an error that is at least that of the most unprecise parameter, which I can't imagine to be more than four significant digits, so doubles should provide plenty of precision. The error introduced by the numerical algorithms can be controlled. I don't see why you'd need fixed point arithmetic.
I guess the GP comment was discussing that, with this new measurement of pi, we now have enough precision (in pi) to reference a point this small on an object this far away. Once you account for all the other uncertainties in referencing that point, as you mentioned, all that precision in one dimension of the measurement is completely meaningless.
It still feels weird that you'd use an arithmetic with guaranteed imprecision in a field like this, but I can definitely see that, as long as you constrain the scales, it's more than enough.
They put in scheduled fix the course burns, as there's a lot of uncertainty outside the math - the fuel burns probably can't be controlled to 5 sig figs, for example. Also, although I have no idea if this matters, N-body orbital mechanics itself is a chaotic system, and there will be times when the math just won't tell you the answer. https://botsin.space/@ThreeBodyBot if you like to see a lot of examples of 3-body orbits. (maybe just in 2d, I'm not sure).
