Why radio exhibits better sensitivity with a wider deviation when Rx bandwidth remains constant
The RF sensitivity is basically the point where the receiver starts to produce a bit error rate that becomes significant or problematic. An FSK receiver (or an FM receiver) converts a frequency deviation (peak to peak) to an analogue voltage (peak to peak). If the basic noise in the receiver is such that it is approaching a similar order to the demodulated analogue peak to peak voltage then you have reached the limit of acceptability for the receiver. 

However, if you increase the deviation, the analogue signal increases and the noise is relatively smaller and less problematic. It's the same argument about using 5 volt logic levels - there is more noise immunity when using 5 volts than 3.3 volts.