Does the "only evidence" count since the source of the evidence is shown to be unreliable as it didn't even know Riker's first name or the name of the ship it claims it knows as its top speed, the Drake?
You can say that Riker's holodeck recreation in TATV was 'unreliable,' but it is still considered to be what happened. So yes. Fake-Rice made a statement about the Drake's top speed, and Riker didn't correct him. So that's the only evidence we have. Feel free to disagree with it, but there it is.
Regarding Warp 3 being the Light Cruiser's alleged top speed... why would a TNG-era light cruiser (that Riker was offered command to) be slower than the NX-01?
It's not. The NX-01's warp five max was the old-old scale (i.e. 100 years before TOS, which was already the old scale a century before TNG.) By TNG, the Drake's warp 3 max could have been warp 7 in the old scale, when the ship was originally built.