Why is FFT a good method for detection of bearing faults? Why can't we do the analysis in time-domain?
Short answer: We can, but it is not always as robust as frequency domain techniques.
Longer answer: This is very broad topic, but let me try to throw some light. Time domain techniques tend to be naive. For example some are based on the RMS energy monitoring of the signal being recorded by an accelerometer. Main disadvantage is that on early stages of fatigue you can't distinguish between faulty and working bearing. Additionally setting a threshold is a real pain. Also some people are using methods based on crest factor (ratio of peak to RMS), knowing that crest factor for impulsive sounds is high. Although there are some 'fancy' methods for analysis in time domain, such as: New Time Domain Method for the Detection of Roller Bearing Defects
Frequency domain techniques (such as Cepstrum or Bispectrum) on the other hand are more robust and have properties that are desirable, such as you can very easily correlate detected fault frequency with a specific bearing and type of damage. For example when you use methods such as Bispectrum, which is (simply speaking) based on detecting presence of higher harmonics. Because it is also taking phase into consideration, in theory even if harmonic is almost covered with noise, you can still detect if its correlated with fundamental frequency - smart isn't it?
An FFT is a computationally efficient way of decomposing certain types of system responses. There may exist roughly equivalent time domain methods of analysis that would require far more computation for similar S/N statistics.