Well you'll always need to consider the over all power the plane can generate and how much can be diverted to radar. You'll also require communication, air purification system, navigational computer and so on.
Basically larger radar does not always equate to larger out put based on how much you can allocate power within a limited system.
As for utilizing received faint signals, better recognition algorithm with faster computing power is also a factor in spotting a target. Without this simply enlarging the radar does not equate to a better radar system in comparing one from the other.
A radar's total signal output is basically the superposition of the individual signal outputs of all the radar antenna elements (i.e. T/R modules) in the radar.
Yes, the bigger the size of the radar in terms of the number of T/R modules, the larger the output power of the radar, and also, the higher the receiving sensitivity of the radar because of the higher number of TR modules to receive the signals.
Of course, this is considering all else being equal, which should be the case in a general sense since the T/R modules basically use the same state-of-the-art technology, i.e. they being GaN based.