Consider 'Hollow Pursuits' [TNG]
Invidium starts screwing up Enterprise systems. The effects are unknown because invidium has not been used by the Federation in over a century, according to Duffy, and most of the systems affected were not invented when the material was in use.
"Who knows what could happen with transporter..."
The year is 2366.
We know that transporters were in widespread use in the 2250s, although they were a bit slow.
We may conclude that invidium went out of use somewhere in the early 23rd century (any earlier and Duffy would have said 200 years). Since the effects of invidium on transporters isn't known to the engineers on the flag ship of the fleet, we can conclude that transporters were first invented around a similar time.
So what the hell is a transporter doing on NX-01?