Question

Is there a quantifiable tradeoff (in terms of effective broadcast distance) between the height of a radio tower and the power of the transmitter?

Answer 1

The power of the transmitter is completely independent of the height of the tower: it depends only on some design features of the antenna: the actual shape, the input current, the impedence etc. Based on geometrical arguments, the power roughly drops like $1/r^2$ (at least in the far field).  The angular distribution of this power does depend on antenna characteristics, i.e. a half-wave dipole does not radiate in the same way as a Hertzian dipole. This will depend on the relative geometry of the antenna and the reception point, v.g. there might be particular orientations where the antenna doesn't radiate for instance.