I think this is right. Any measurement will have finite precision, so while we might be able to discover some maximum precision that the universe uses eventually, we won't ever be able to prove that the universe has infinite precision representations from finite precision measurements.
Only so long as we use the rationals as an approximation. If we expect them to be exact then they are as bad as the integers.
The continuum is the reality that we have to hold to. Not the continuum in the Cantor sense, but in the intuitionalist or constructivist sense, which is continuously varying numbers that can be approximated as necessary.