"Velocities add" is also mechanics 101, and turned out to be incorrect at sufficiently large speeds. I think we have to go to Noether's theorem to show why we should be so sure the EM drive doesn't work: EM drive proponents must either tell us where the momentum goes, or explain why space isn't symmetric under translation.
(Actually, fun side note, in a gravitational field, space isn't symmetric under translation. IIRC, this was the original problem that caused Noether to prove her famous theorem. She found that while there was still a conserved quantity of a sort due to "translations" in space and time, it wasn't just the energy-momentum tensor, but also depended on the gravitational field. In any case, the EM drive is supposed to work in flat space, so that shouldn't become an issue.)
(Actually, fun side note, in a gravitational field, space isn't symmetric under translation. IIRC, this was the original problem that caused Noether to prove her famous theorem. She found that while there was still a conserved quantity of a sort due to "translations" in space and time, it wasn't just the energy-momentum tensor, but also depended on the gravitational field. In any case, the EM drive is supposed to work in flat space, so that shouldn't become an issue.)