In the discussion below, when frequency is mentioned, it refers to the complex frequency s = σ + jω, not the frequency of the PLL.
Technically, the low-pass filter is the controller in a negative feedback loop. This controller shapes the frequency response magnitude of the closed loop such that it has high gain at low frequencies to reject low frequency disturbances (and track changing setpoints, if that is desired) and has low gain at high frequencies to reject high frequency noise. Often, any filter that has a high gain at low frequency and low gain at high frequency will change the loop shape in this fashion. All of the filters you mentioned can do that (provided that what you call lag-lead - but most call a lag controller - has the pole at a much lower frequency than the zero).
Plot the Bode magnitude plots for each - they aren't that different. The single pole active RC filter has a flat low-frequency response at a non-infinite positive dB magnitude, then after the pole is reached, its magnitude drops 20 dB/decade as frequency increases and it goes to negative infinite gain at infinite frequency. The lag controller has a flat low-frequency response at a non-infinite magnitude, then after the pole is reached, its magnitude drops 20 dB/decade as frequency increases until the zero is reached, at which point the frequency response levels off instead of going to negative infinite gain at infinite frequency. A PI controller has infinite gain at low frequency due to the pole at the origin of the s-plane. It's magnitude drops 20 dB/decade as frequency increases until the zero is reached, at which point the frequency response levels off. The PI controller is nothing more than a lag controller whose low-frequency pole has moved to the origin of the s-plane.
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