The 200 year reference has to be fudged. Folks can say "200 years" or "two centuries" and still not mean exactly that amount of time. In WNMHGB Kirk first says in his log the Valiant
went missing over two centuries ago and then later in another log entry he says nearly
two centuries ago. So which is it or did he get more updated information later? My conjecture is the Valiant
was one of those fast STL ships that had an early space warp drive fitted to it then went missing not long thereafter.
Also in the early years of warp flight it might not have worked the way it does in the Pike, Kirk and Picard eras. Being able to go FTL even if not much above light (say maybe 10-20%) would be revolutionary and a helluva advancement, but it's still pretty slow going even to the nearest stars. And maybe warp flight in the early days is more of a point-to-point affair and navigating while at warp is real tricky. Also sustaining warp flight might also be tricky, sustaining a stable warp field particularly while in something like a magnetic storm might have been considered hazardous. In that context then Kirk's assumption that "the old impulse engines weren't strong enough" might have more meaning.
Of course it's all conjecture, but it's a way to stitch things together piece by piece.
I'll add one more thing, and yes it contradicts ENT. In "The Cage" Jose Tyler comments on how much faster the then newer ships are and thereby implying ships of the Columbia's
era were much slower. In "Balance Of Terror" Spock mentions the ships of a century earlier were "primitive." And later in "The Ultimate Computer" we learn Daystrom's duotronic computer systems revolutionized starship design and capabilities some twenty-five years earlier. Hmm, what exactly happened 25-30 some years before the Kirk era? Maybe what happened was that Daystrom's duotronic computer systems were so much faster than what came before that it lead to a technological leap, particularly in terms of warp flight, that made everything before seem comparatively primitive. It would be akin to what computer technology has done for us over the past 25-30 years. Daystrom's systems possibly now have made navigating while at warp a lot easier and safer. Daystrom's systems might have solved sustainability problems for stable warp speeds. Now instead of maximum Warp 2 and 3 with occasional risky bursts of 4 you can easily reach Warp 5-8 and hold it and also not worry about piling into a planet or something because you can clearly see where you're going and easily steer at that speed as well because your fancy new duotronic systems can make the necessary calcutions and changes so much faster than before. It would be similar to what computer management systems have done for automobile engine performance as well as jet plane performance.
Part of why I have issue with ENT is because they chose to make everything look so neat-and-tidy and really not much different all from what we had already seen taking place centuries later.
Something to think about.