That landing distance for different jets is typically similar if they are landing at the same SPEED (not weight) is no coincidence.
And is in fact quite a simple concept.
With the PRIMARY retarding force being brakes (in this case), the decelerating force from them F = mu * Weight...
Actually, the lack of coincidence is primarily because current autobrake systems are programmed in most settings (other than "Max" or equivalent) to yield a specific deceleration rate, regardless of other conditions (unless outside influences like reversers already exceed that deceleration rate). Hence, the brakes are modulated to keep the deceleration constant, and "published landing distances" for normal landings are pure functions of landing speed.
The equation you cite may be valid, but it does NOT explain the similarity in landing distances as you represent it.
I did just a bit more ciphering in the 747-400 tables and came up with the following:
Base airplane is 250,000 Kg gross weight, Vref = 143 KIAS, max manual braking, antiskid on.
At Vref, landing distance is 4250'
At Vref + 20 KIAS (163 KIAS), landing distance is 5200'
For a 320,000 Kg gross weight, Vref = 163 KIAS and landing distance is 5770' for the SAME CONDITIONS -- a difference of 10%.
So, your assertion that landing speed is the only variable is incorrect in reality. If such a significant difference is apparent for the SAME drag/brake/wheel/tire configuration, there is likely to be a significant difference among airplanes with different parameters.