I always felt like while the Enterprise-B made its own legacy, there was probably a dark cloud looming over her considering that by a lot of people's point of view (In-universe and OOU) the Enterprise-B was the ship that killed James Kirk. and a lot of people probably stopped and went "Oh hey there's Captain Harriman, he killed James Kirk."
If you were talking about people today, yes... in Trek, I don't see why this would be the case.
People in the Federation would likely be more exposed to basic methods of science, capable of critial thinking and applying proper context to the situation (plus the news would have been reporting what actually happened).
The ENT-B was responsible for rescuing refugee El-Aurian's... and given the amount of systems it didn't have at the time, the crew had to get creative (but in all honesty, the ship shouldn't have been going anywhere, even as part of a simple ride around the block without all systems in place).
Offering the reporters a tour is one thing, but leaving drydock without all systems installed.. nope.
Harriman DID offer to go down and reconfigure the deflector, but Kirk decided to go in his stead as he said to Harriman the captain's place is on the bridge.
Problem is, many ships didn't make the cut by being displayed in the Fleet Museum... and that could be because they were destroyed or lost... but received honorable mention in the Museum anyway (like the ENT-C).