PLL startup is its own specialty. A loop amp which heads for the weeds
and stays there may have to have a startup circuit overlaid. You may
see "kickers", "sweepers" or even just a "switch to center" to get the loop
into sane center-seeking range.
Now in the end you'd be wanting to verify that these "patches" do the
job every time, every corner, every in-tolerance external component at
its margins.
But for the core, seeing what happens "bare" leads to "candidate mitigations"
leads to "real thing" and then verification of it all against the universe of
uses and abuses.