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.