Where are you getting 1.41x? What you'd really like to increase is the SNR. As you open up the bandwidth, the amount of energy you can collect in your band increases, but there's no way to collect the energy from only the signal and not collect the energy from noise. So as you increase your bandwidth, your SNR stays the same.

Not all noise is gaussian. And the fact that the noise is random while the signal is not, is useful when you can average and drop your noise floor. But you need multiple measurements to do that.

1.41x is sqrt(2), which suggests that they meant noise amplitude rather than noise power.

Noise power increases twice but signal power increases 4x. Noise amplitude increases sqrt(2) times, signal amplitude increases 2x.

Noise is not gaussian.