Usually superconductor resistance is measured using four probes: current goes through the two outer probes, and voltage is measured across the two inner probes. Then V=IR.

Lead resistance can play a role, but superconductivity is a phase transition: there's a significant discontinuity. It's not a matter of something going from 0 + e (for small e) to 0 (which doesn't happen), but one of going from x (x >>> 0) to 0. When that phase transition happens, it's obvious.

Noob question: how do you know you're not going from x >>> 0 to 0 + e?

Not clear to me how you know the resistance is nothing at all vs just very small.