Earth apparently did mass-build certain types of starship - the indigenous tramps that allowed it to create all-important commercial connections. But it might not have been worthwhile to produce substandard, warp three explorers that could not hope to explore anything beyond the already Vulcan-controlled volume of space within a reasonable time, nor to construct warships that couldn't put even a small dent in the enemy armor. Those had to wait for technological breakthroughs, or for the obtaining of tech secrets by other means such as trade or espionage.
Nevertheless, there's no indication Earth would have hesitated from building starships in general. The Valiant ventured to very deep space just a couple of years after Cochrane's invention, and there was no indication that she would have been a unique vessel - had she been the only explorer from that era, she'd have been far more famous and more immediately identifiable in "Where No Man Has Gone Before".
TAS offers other examples of starships built early on. And ENT adds the early colonization efforts, beginning with the Conestoga. If anything, Earth adopted warp drive ridiculously fast, basically flying 747s across the Atlantic within a decade of the first hop at Kitty Hawk...
On the issue of the NX Project being the first to break warp two for mankind, this wasn't really established. All we learned was that the NX Project broke warp two for the NX Project; many a rocket project today starts with a breakthrough in achieving liftoff, then proceeds through the further breakthrough of achieving a certain height or a certain payload weight, all despite the Apollo program already having broken all the records a long time ago. Earth might well have had plenty of starships capable of warp three when the NX Project was started; it's just that none of those could evolve into warp five starships, not without the help of the NX Project.
Timo Saloniemi