In "reality", there is no such thing as an electron that has some mass. It's leaky abstractions and approximations all the way down. If we had a "complete" theory of physics, it would still be the same situation for all practical purposes, since you can never know "for sure" that is it "correct".
Hence, rationals are plenty sufficient for all of physics, assuming you bother to reformulate all the theories that are historically based on real numbers. However, there is no point in doing this, so only a few niche people try (who, of course, do it for philosophical/aesthetic reasons).
The tools of mathematics are actually just better suited to working on the continuum than on discrete numbers. So maybe that does suggest that when physicists use the tools of mathematics they will always wind up with real approximations to discrete reality.
No matter that physical reality insists that the number of people in a population or U-238 atoms in a lump of metal has to be an integer, mathematics will tell you that that integer nonetheless has a distinct, nonalgebraic natural logarithm, and that that number is useful in predicting how many people will be in that population or U-238 atoms will remain in that lump of metal at some later time - even if the raw result of any such calculation will be a nonsensical noninteger.
Hence, rationals are plenty sufficient for all of physics, assuming you bother to reformulate all the theories that are historically based on real numbers. However, there is no point in doing this, so only a few niche people try (who, of course, do it for philosophical/aesthetic reasons).