What you say is true as far as that goes; if it were only a case of getting precision in observed physical quantities proportional to the precision of our measurements, I wouldn't be so bothered about it.

But things like the Born probabilities on electron spin measurements, or the gamma ray observations, suggest that a small number of bits being given to us as measurements are backed by much higher precision calculations behind the scenes, with no obvious way to put an upper bound on that precision.

In particular, we had for a long time been thinking of the Planck scale as the likely upper bound, but that now appears to be broken, and I'm not aware of any other natural/likely bound short of infinity.