I don’t know how old the stirrup idea actually is... BUT I thought that the innovation didn’t reach Europe until after the first sack of Rome. Was it 437 AD?
Despite the fact that the Romans studied everyone and everything they came across, and did an excellent job adopting and improving things, in the armed forces they were just hidebound enough not to start using the stirrup, and of course, they still had the auxiliary cavalry as well as Roman regular cavalry. By the time of Trajan, the Roman regular army was itself made up of non-Romans who were nonetheless citizens of the Empire.
It's also worth considering the possibility that the Roman cavalry *did* use the stirrup and that the idea that they never used it arose as a sort of just-so faux explanation, perhaps by Gibbon.