You are in the right direction. But the scaling in not linear in all the cases. C depends on the area and the thickness of the dielectric. The thickness is not changed much but the area decreases as a square of the linear scaling parameter.
Just a rough example: consider the linear scaling is by a factor of 2.
C will decrease by a factor of 4. For the same power, you can increase f by a factor of 4.
However, with linear scaling by a factor of 2, the power cannot be kept constant. But smaller size has better power dissipation (greater surface to volume)- the actual dependence is rather complex.
Actually power consumption will decrease by a factor of 2. Then the frequency will actually increase by a factor 2 (not for) but the details are approximate.
Power consumption will depend also on frequency, voltage and capacitance. It is better to reduce the voltage because that is quadratic term,