Sinusoidal signal as density
For a sinusoidal signal, noise and bandwidth do not fit in the
  context for me.


Nor for anybody else. Since a pure sine wave has zero bandwidth, the signal power over any finite bandwidth is independent of bandwith. That is, a 1 watt signal will have a 1 watt power for any non-zero measurement bandwidth. 

So root-hz simply doesn't apply.

As you have realized, it works fine for noise, where the amount of signal varies with bandwidth.