Whether or not we want to use smoothed data (like by taking a 3-month average) or nearly instantaneous (such as daily) data depends on the purpose of your analysis. In the case of ENSO, which is a slowly evolving climate phenomenon, it's advantageous to average out some of the daily and weekly noise of SST variability in the Nino 3.4 region that is unrelated to ENSO (and instead may be related to other phenomena like tropical instability waves). So, in this case, the 3-month-average filter applied with the ONI helps to bring out the ENSO signal more cleanly from the noise of unrelated variability. If, however, you were interested in some weather or biological phenomenon that responds to instantaneous Nino 3.4 region SSTs, whether or not it's tied to ENSO, then it would be better not to use the ONI and instead to use the daily or monthly Nino 3.4 SST. Like with so many applications, the best form of the data depends on purpose.