I think this is based on some false idea that error is somehow “not theoretical” which makes no sense. There is a rich set of about error and even an entire field dedicated to its study (statistics). The only reason we are interested in theories in the first place is because the theories correspond to the real world. If you have some theory about time which cannot be measured, or cannot explain errors, then that theory belongs in the garbage. (Think about this: if your theory cannot explain errors, but any actual experiment produces errors, then any actual experiment will demonstrate that your theory is “wrong”.)

Or consider other concepts like “flat”. What does it mean for a surface to be “flat”? No matter how flat a real surface is, it will never be mathematically flat. So either we have to throw away the word “flat” and stop using it outside mathematics (which is a stupid idea), or we define “flat” to mean that a surface is within some deviation from mathematical flatness.

You can’t escape error, even in theory, unless you leave the world of physics and do pure mathematics.