Rendezvous and docking are not synonymous. Aldrin did his Ph.D on orbital rendezvous, and was known to talk about the subject endlessly. IIRC during his Gemini mission when their computer malfunctioned he got out paper charts and made the calculations himself.
But docking is the actual joining of the the two spacecraft that were brought into the same orbit via rendezvous. Armstrong performed the first docking during his Gemini mission, although things went awry shortly after (not his fault, and he pretty much saved them, but that’s another story).
As for the LMP actually flying the LM, that happened once. On Apollo 12 the commander, Pete Conrad, invited Al Bean the LMP to fly it while they were behind the moon and out of radio contact. This was not part of the plan, and according to Bean a delightful experience only made possible by Conrad’s indulgence.
FTR, Wiki cites–and I seem to recall either entering this myself or taking part in the discussion in the Aldrin entry–Andrew Chaiken A Man on the Moon (Penguin, 1998) as presenting evidence of the decision making of NASA on this topic.
Aldrin’s PhD thesis was about space rendezvous methods and he was an expert. However this level of academic knowledge was not required for the mission.
The LM was the first digital fly-by-wire aerospace vehicle, and had completely redundant navigation and guidance computers. Unlike most fly-by-wire airliners it also had a completely manual reversion mode including a non-computerized analog pathway from the hand controller to separate redundant solenoid propellent actuators on each RCS thruster. Activating this required only a full deflection “hard over” on the hand controller to trigger the alternate path.
There was a practiced contingency procedure to ascend from the lunar surface and achieve orbit using no computers and no inertial platform whatsoever. It involved using charts, a stopwatch and making stepped pitch changes to align window etchings with the lunar horizon. Astronaut Gene Cernan said he achieved this in the simulator and felt it was possible. In this contingency the Command/Service module would take the lead in rendezvous after the LM achieved orbit.
All Lunar Module Pilots were fully qualified to fly the vehicle. In his autobiography Apollo 17 commander Cernan said his LMP Harrison Schmitt was more proficient on the Abort Guidance System (AGS) than he himself was.
Armstrong’s cool handling of, and impeccable timing during, the May 1968 “Flying Bedstead” crash also impressed NASA - see the third paragraph here: Neil Armstrong - Wikipedia
