The direction to the source is normally combined with symbology indicating the likely purpose of the radar – Airborne early warning and control, surface-to-air missile, etc.
Agile radars like AESA (or PESA) can change their frequency with every pulse (except when using doppler filtering), and generally do so using a random sequence, integrating over time does not help pull the signal out of the background noise.
Modern RWRs must be made highly sensitive (small angles and bandwidths for individual antennas, low transmission loss and noise)[1] and add successive pulses through time-frequency processing to achieve useful detection rates.
[2] Ways of reducing the profile of a radar include using wider bandwidth (wideband, Ultra-wideband), frequency hopping, using FMCW, and using only the minimum power required for the task.
Modern phased-array radars not only control their side lobes, they also use very thin, fast-moving beams of energy in complicated search patterns.
In addition to stealth considerations, reducing side and back lobes is desirable as it makes the radar more difficult to characterise.