The Saturn V (Apollo moon rocket launches) were somewhat different from the Space Shuttle, with very gradual initial acceleration (seen in contemporary footage). Most of the thrust is used to overcome the sheer weight of the rocket, and only once the propellant begins to burn off, does this lead to increasing acceleration (given a constant thrust). This process might be part of the 'myth' that you quote.
The Wikipedia entry for
Saturn V rocket data indicates initial acceleration (my calculation) of 2m/s^2 vs 5m/s^2 for the
Shuttle.
There is a nice video showing altitude and velocity vs time
https://www.youtube.com/watch?v=F0Yd-GxJ_QM for the entire first stage Saturn V burn (less than 3 minutes total), which consumes more than 4/5 of the rocket's launch weight. Just before stage cutoff, the acceleration is around 20 times that at liftoff. The second stage then goes though another period of gradual acceleration as its propellant burns off.