I found the answer. I built the same circuit in a simulator and watched its behaviour. The critical missing piece was the behaviour of the signal being fed into the VCO.
I had modelled the charge pump as a floating point value that increased when UP was set, and decreased when DOWN was set, at a predefined rate (dV/dt), or doing nothing when neither or both ...
Are the PI and IIR LPF filters equivalent?
A PI filter is an IIR LPF, but a low-pass filter is more general. So -- no.
Is my understanding of this "modified" PLL structure correct?
I can't tell, but I think you're lacking in the understanding of closed-loop control systems.
Should the IIR filter work in this case?
Maybe, but it depends on ...