This is true only if (1) one signal is very quiet, and (2) you're using a SSB mixer or a zero-Hz IF, so there's no sideband folding.
To see (1), consider multiplying a signal by itself, as in an AM detector. One can't replace the RF and IF filters in an AM radio by an audio filter--after all the intermodulation, it's too late.
To see (2), consider image rejection in a superhet. Ordinary mixing makes components at F_rf+F_lo and F_rf-F_lo land on top of each other. Even if there's no signal at the image frequency, if you leave out the filter you lose 3 dB in SNR due to the noise at the image.
The PLL is going to have most of what you'd need for a mixer/filter measurement already in it--a phase detector and lowpass filter, plus probably a bandpass filter to allow the loop to acquire lock in the first place (there are other methods, but this is a common one).
The BPF/mixer/LPF approach is good if the phase difference you need to track is small, the signal amplitudes are stable, and you know pretty well what it is already (e.g. that the relative phase is always 0.1 to
0.4 cycles). Otherwise it's a mess--the slope of Vout vs phase will depend on the amplitude, and you'll only get the phase modulo pi radians anyway.You can use a limiting amplifier ahead of the mixer, which fixes the amplitude dependence problem, but will inevitably cause some AM-PM conversion--the amplitude noise will turn into phase noise.
Using a PLL to clean up the noisy signal has two advantages here: first, it gives you the constant amplitude you need without needing a limiter, so you potentially have better phase accuracy; and second, you can use a frequency divider in the loop to allow you to track over a wider range in phase (many cycles, potentially). The PLL's low AM-PM conversion arises from running the phase detector at null (zero volts dc output)--changes in signal amplitude cause changes in slope (and hence in loop gain) but since zero times anything is still zero, amplitude changes ideally don't cause any phase error at all.
There's a phase ambiguity in loops with dividers, because when you turn the box on, the counter will wake up in some state M, that won't in general be the correct relative phase.
The other disadvantage of PLLs, as others have pointed out, is that noise and loop dynamics cause the PLL output to have additional phase variations. This is generally quite benign if the PLL is properly designed, by which I mean that the loop is properly frequency-compensated and the input signal to the phase detector doesn't contain large components with offset frequencies >~ the loop bandwidth.
Another approach that has many of the same advantages is to make a phase-tracking loop with a phase shifter instead of an oscillator, and use its output directly. I did a digitizing version of this in grad school, which worked great.
Cheers,
Phil Hobbs