Why is there no reference in the formula to the physical characteristics of the aircraft?
In level steady state flight Weight = Total Lift = Vertical Lift. The ratio of Total Lift to Vertical lift = 1. In a bank more lift must be generated to maintain sufficient vertical lift. The ratio of Vertical Lift to Horizontal Lift is Tangent (bank angle). The horizontal lift component is what changes the direction of the flight path. The Force moving it in a new direction is mass x acceleration. This is always proportional the lifting force regardless of weight.
As lateral motion accelerates from zero and direction is always changing, the side drag of the aircraft is not nearly as significant as the sideways acceleration compared to the energy of forward motion.
Here the "physical characteristics of the aircraft" are accounted for in Velocity: the sailplane has a much tighter turning radius than the jet airliner because of a much lower stall stall speed.
This is where airfoil type and wing loading come into play.
The key to understanding this is to put mass "m" back into the equation:
R = mv$^2$/mg (tan bank angle)
We can now clearly see the energy of the object is proportional to the square of the velocity. Weight (therefor lift) is mg. One simply gets a slower rate of turn with
a faster moving object for a given amount of sideways acceleration.
Now, if one cuts power and wants to turn, physical characteristics of the aircraft will matter. Broad, low aspect, draggy wings make biplanes more maneuverable in turns than their faster mono plane counterparts. This is one reason triplanes were all the rage before new tactics were developed.