why do these two different mechanisms produce the same time dilation factor in this scenario?
Well, you are making an assumption here, and I recommend you let go of that assumption. Do not assume something about mechanism; leave that open.
Crucially: GR subsumes SR.
GR content and SR content do not live side-by-side; GR covers the scope of SR, and beyond.
A specific aspect of that: in terms of GR: local measurement cannot distinguish between velocity time dilation and gravitational time dilation - as a matter of principle.
There is the thought demonstration of a wheel-shaped space station where the space station is rotating for the purpose of pulling G's.
Levels of the space station that are further away from the axis of rotation have a larger velocity than levels of the space station that are closer to the axis of rotation. Correspondingly, for different levels there will be a difference in the amount of proper time that elapses.
In terms of GR:
In accordance with the principle of equivalence: GR facilitates description of the time effect in terms of gravitational potential.
Throughout all levels there is a gradient in gravitational potential. From level to level a different amount of proper time elapses, in accordance with difference of gravitational potential.
As a matter of principle:
Locally no measurement is available that would allow you to distinguish between velocity time dilation and gravitational time dilation.
Distinction between velocity time dilation and gravitational time dilation is not inherent in the phenomena. (There is a bit of a parallel with the principle of SR here: the magnet and coil thought experiment: making a distinction between the magnet as the moving part, or the coil as the moving part, is not inherent in the phenomena.)
Unfortunately, in introductions of GR as successor of SR it is common practice to treat gravitational time dilation as something that occurs in addition to velocity time dilation. (In approximate calculation the two can be treated as separate, and additive.) The thing is: it's better to emphasize from the start: GR subsumes SR, in the most profound sense.
There is another context where making a distinction between velocity time dilation and gravitational time dilation leads away from understanding: the amount of proper time that elapses for clocks onboard a satellite in orbital motion around its primary.
In the case of the Earth:
In low Earth orbit: for clocks onboard a satellite in low Earth orbit a smaller amount of proper time elapses than for clocks on the surface of the Earth.
The constellation of GPS satellites orbits at an altitude of about 20,000 kilometer. For clocks onboard those satellites a larger amount of proper time elapses than for clocks on the surface of the Earth.
There is in fact an orbital altitude such that the amount of proper time that elapses for clocks onboard satellites at that orbital altitude comes out the same as for clocks on the surface of the Earth.
That case is discussed by Kevin Brown, in his resource 'Reflections on Relativity': Proper time in circular orbits
For equal amount of proper time elapsing Kevin Brown arrives at an orbital altitude of about 3000 kilometer.
There is another example:
Worldwide: for all clocks located at sea level the same amount of proper time elapses.
Now, a clock located at the Equator is circumnavigating the Earth's axis at about 1500 kilometers per hour. So if were to count velocity time dilation and gravitational time dilation as distinct then you would not necessarily expect that for a clock on the Equator the same amount of proper time elapses as for a clock located at the pole.
About the Earth's rotation:
The rotating Earth is in hydrostatic equilibrium. There is an equatorial bulge; the equatorial Earth radius is about 21 kilometer larger than the polar radius.
About the state of equilibrium: it is a balance of two energies.
The extra height of the equatorial region is a reservoir of gravitational potential energy; if there would be opportunity for the Earth to contract to a perfect sphere then there would be release of gravitational potential energy. The Earth started out as a protoplanetary disk, and as that disk contracted towards spherical shape release of gravitational potential energy provided the energy to increase the rotation rate of the proto-Earth. (When a rotating system contracts the angular velocity increases, such that the angular momentum is conserved.) As long as over the course of contraction the rate of release of gravitational potential energy exceeds the rate of increase of kinetic energy the contraction continues. The contraction process has halted at the point where those two rates of change arrived at equilibrium.
Once arrived at the equilibrium shape there is no more opportunity to dissipate energy. There is a lot of energy in the system, but no opportunity to dissipate any of that energy. We can think of the equilibrium shape as a state of lowest possible total energy.
In the case of the Earth's gravity and the Earth's rotation: the gravitational effect is at a level of weakness such that the spatial effect of spacetime curvature is negligable as compared to the time effect of spacetime curvature.
Then in terms of GR the equilibrium shape of the Earth (equatorial bulge) can be thought of as a time equilibrium. A rotating celestial body will gravitationally contract to a shape such that everywhere over its surface (same geopotential height) the same amount of proper time elapses.
