IIRC, "First Flight" established they would have had a top speed of warp 1.9.
What "First Flight" did was to establish that reaching warp two with the new engine on the NX-Alpha test rig was a major breakthrough in the development of a warp five engine.
This doesn't exactly mean that other types of engine would not have already achieved, say, warp three or warp four. Their respective histories may also have involved the use of test rigs that each achieved warp two in turn, and thus marked a major milestone in the respective engine development programs.
Sure, the flight in "First Flight" has a Chuck Yeager feel to it. But that's somewhat undermined by the fact that our Yeager substitute very well knows that everybody else in the galaxy is already doing much better than warp two, and that ships capable of warp seven or so have rather frequently visited Earth, bringing diplomatic mail to the Vulcan Embassy, and apparently also carrying human passengers to Vulcan for its Earth Embassy... Reaching warp two is not a giant leap for mankind. It's just a giant leap for the Henry Archer engine development team.
As for the supposedly older ships we see, they aren't noticeably smaller than the Enterprise
. They also seem to feature very potent armaments. Several possibilities exist, but it seems probable that Starfleet before ENT was a fighting force, unable to explore anything because the ships lacked the range and speed required to reach unexplored space. The older fighting ships may have featured heavy phase cannon armament for decades already, and NX-01 only lacked it because its installation was delayed by the hasty launch in "Broken Bow". Or then phase cannon were a new invention, and NX-01 was among the first ships to have it installed, but all the older ones had it retrofitted within the year, too (a fairly trivial task, considering our heroes could do it all on their own in the middle of nowhere).
We never learn anything truly interesting about the past exploits of the United Earth Starfleet. It certainly didn't appear to strike fear in the hearts of space pirates... But perhaps it had fought a few easy wars against weak opponents, just to get some respect. Or perhaps it had defeated minor attacks against Earth (say, the Kzinti ones mentioned in TAS "The Slaver Weapon"), while more powerful enemies were discouraged from challenging mankind, by its powerful Vulcan protectors.