Depends. For most riders, the engine really is the dominant factor. Most of the bits you can hang on the bike really won't make a big difference. So if you're struggling to hit 15mph on your hybrid, odds are you'll be struggling just as much on a more specialized bike.

The point where the bike makes a big difference is when the rider can do a big upgrade on the bike, and it's gotten hard to upgrade the engine. My partner's mom and dad helped me gang up on him this year, so he has a new bike. Old mountain bike was about 45lbs, and he was riding it as a single speed, regularly cruising at 15-17mph. Average speed, so including stops. New bike is about 25lbs, is a real single speed, and it fits well. He gained a good 3-5mph average speed from the upgrade. (that's *with* the new bike being geared considerably lower)

We spent about two years hashing out what my partner needed from a bike, puzzling over fit, and just plain thinking. And all the while, he just kept putting miles on his mountain bike working on building speed and skill. At the beginning of those two years, he took a test ride on a bike. Came back swearing it was too slow, and was not what he wanted because it was a single speed. Two years later... that exact bike is the one we ended up buying.