Does the bandwidth at the receiver really affects the noise in digital modulations?
Does the bandwidth at the receiver really affects the noise in digital modulations?

Yes; noise is white, i.e. it has constant power spectral density, and cutting out a smaller part of that that spectrum means lower power (by the definition of density!).
What you forget to mention in context is that the formula is normalized to the symbol rate and thus bandwidth of your system; that means that while increasing the bandwidth does increase the observed noise power, the amount of noise energy per symbol is unaffected.