Why did Armstrong pilot the LM, when Aldrin was tasked as Lunar Module Pilot?

Buzz Aldrin is credited (or was tasked) with being the Apollo 11 Lunar Module pilot. However in fact it was Neil Armstrong who piloted the craft down when the LM guidance computer overflowed, with Aldrin reading out to navigation data.

Why didn't Aldrin, the LM pilot, "take the stick" when human intervention became necessary?

EDIT: Even though the question has been answered, I just found this mention that also on Apollo 12 was the LM piloted not by the pilot, but rather by the commander.

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The question should really be, why did Armstrong pilot the Lunar Module (LM) manually, and the autopilot was switched off. According to Tales From The Lunar Module Guidance Computer by Don Eyles, Armstrong and Aldrin never switched places during the LM powered descent and landing phase. Armstrong was on the left side of the LM cockpit where the manual controls were, and Aldrin on the right and responsible for working the DSKY (Lunar Module Display and Keyboard Unit), essentially instructing the flight computer.

For reference, here's a photo of the Apollo Lunar Module's interior (source and credit: NASA HQ):

It seems that the 1202 Program Alarm that the LM was experiencing prior to the landing prompted the switch from AUTO to ATT HOLD flight mode. From mentioned source:

ATT HOLD meant the digital autopilot's Rate-Command Attitude-Hold mode, in which the astronaut could command an attitude rate by deflecting a joystick. After the stick was released the autopilot nulled rates to maintain the present attitude.

Armstrong, as the mission commander, had to make a decision whether to proceed with the landing or not, while dealing with it with the mission control in Houston. It turned out that:

Due to an error in the checklist manual, the rendezvous radar switch was placed in the wrong position. This caused it to send erroneous signals to the computer. The result was that the computer was being asked to perform all of its normal functions for landing while receiving an extra load of spurious data which used up 15% of its time. The computer (or rather the software in it) was smart enough to recognize that it was being asked to perform more tasks than it should be performing. It then sent out an alarm, which meant to the astronaut, I'm overloaded with more tasks than I should be doing at this time and I'm going to keep only the more important tasks; i.e., the ones needed for landing...

— Margaret Hamilton, lead Apollo flight software designer

And the computer was prompting Armstrong and Aldrin with other alarms, for all of which they received a go to proceed from Houston. According to Human and Machine in Spaceﬂight, David A. Mindell, 2008 Massachusetts Institute of Technology (PDF):

With the push of a button, the astronaut could abort the landing, an action practiced countless times in simulation. Yet he held off. Armstrong later explained himself as a mechanism:

‘‘In simulations we have a large number of failures and we are usually spring-loaded to the abort position. And in this case in the real ﬂight, we are spring-loaded to the land position.’’

Houston checked out the problem. Young engineers recognized it from a recent simulation, and conferred with their support teams in the back room. They quickly found the cause. The computer was overloading and restarting but not shutting down. It was ignoring low-priority tasks, but these were not critical for the mission. ‘‘We're go on that alarm,’’ the ground controller replied, meaning the LM could proceed. For his role in clearing the landing, engineer Steven Bales later accepted a presidential award on behalf of the ﬂight control team.

Armstrong surveyed the computer display; it had frozen. He checked the LM’s systems. The vehicle seemed to be responding to his commands, meaning the computer was still running. So he continued.

Emphasis mine. So it seems that the switch to manual was due to a set of circumstances prior to the landing. Armstrong, as the mission commander, continued piloting the LM in ATT HOLD flight mode after the 1202 Program Alarm, and all subsequent alarms they were in contact with Houston about. After mission control gave a go on the alarms, and having the control over the flight computer, he made a command decision and took The Eagle down to its landing spot — Tranquility Base at the Sea of Tranquility (Mare Tranquillitatis), the Moon.

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My understanding is that landing was Armstrong's job from the start. It would have been Armstrong rather than Aldrin who landed the Eagle even if nothing had gone wrong. The same has been true on other firsts. It's the commander who does it the first time (e.g., first docking in the Gemini program, first landing in the Shuttle program). It becomes the pilot's job only after it gets mundane. –  David Hammen Apr 21 at 17:42
Thank you TildalWave. I'll wait a bit for David or someone else to add a bit more info before I choose an answer. –  dotancohen Apr 21 at 19:56
@dotancohen We've discussed this a bit with David in our Space Exploration Chat. I've just added a few clarifications in my answer, hopefully that improves it. The point is that Aldrin was never meant to pilot LM in manual mode, and he was at the right side of the LM from the start, while Armstrong was at manual control joysticks and all at his station to the left. If all went well, then Aldrin would have control of the flight computer. As it didn't, the commander had to manually pilot LM while Aldrin assisted him. I agree with David, anomalous readings do call for commander to take over control of the ship. –  TildalWave Apr 21 at 20:02
Either case, there's no need to hurry with accepts or alike. We'll get to the bottom of it. That's what peer review is all about. ;) Also, here's a nice infographic on Apollo 11 landing that I found during my search for relevant citations. –  TildalWave Apr 21 at 20:04
Thanks. After quite a bit of reading I have found some confirmation of David's assessment that safety-critical 'firsts' are done by the commander. Therefore, despite the titles, a manual landing was Armstrong's responsibility. It can be seen here that Young, not Crippen, landed STS-1, for example. –  dotancohen Apr 23 at 6:40