Before the war, humans (as well as Vulcan, Tellarites and Andorians) must have warp drive, otherwise they wouldn't know each other or be a danger to the Romulans.
After the war, these four races found the Federation, implement defences among the neutral zone and explore space, so they use warp drive.
Spock's lines suggest that in between, i.e. during the war, technology is simpler. Yet humans have warp capability. So it has to be a free choice of a simpler technology, e.g. impulse ships with H-bombs instead of warp ships with photon torpedoes. Why?
The only reason I can imagine is that the war is, as you described, pretty ugly and bloody such that cheaper/faster to produce ships and weapons are chosen.