Indeed, that part of the analysis was swept under the "long story short" about the Fermi Paradox & The Drake Equation.

So, having studied this quite a bit, it wouldn't occur quite as you envision. However, the net effect of what could happen would effectively be the same as the result you have imagined.

If we were to become significantly more technologically advanced, and, saturated the EM spectrum as such, this itself would be a signature of technological life. It would not, by chance, match the background galaxy radiation field.

The above analysis takes into account the possibility that technological civilizations might only have a narrow window of being detectable, of say, a hundred years to a few hundred years (after which time, they could still be there, but undetectable).

Under this possibility, the aliens would be making a conscious decision to become undetectable (out of fear of other alien races), or, by some fluke chance, have built technology which emits less EM waves.