Assuming our lungs are like our engines
They are not. Engine (at least spark-ignited reciprocating one) uses most oxygen in each charge and its RPM is limited, so it can't be run faster to compensate for the lower air density.
However, our bodies only use small fraction of oxygen from each breath and we normally breathe much slower than we can if need be.
They are not. Wings depend on dynamic pressure, but dynamic pressure is not involved in any significant process in the lungs.
and dependent on density altitude
Well, they don't. The limiting factor for respiration is the rate of association of oxygen to the hemoglobin. That depends primarily on partial pressure of oxygen and since partial pressure is just pressure times volume fraction, on pressure altitude, not density one.
Temperature does affect the association rate, but since the temperature is that of blood, and we are warm-blooded, the temperature is fairly constant. And the effect is different from that on density anyway.
There are many other factors that affect the association rate that are physiological and have nothing to do with the weather and everything to do with the person.
Therefore there must be safety margin built into the rules. I can offer two anecdotes:
When I went up to 14,000 ft, we spent several days around 12,500 ft. Still, just around 14,000 ft, one of the 16 people there got, rather suddenly, symptoms of altitude sickness and had to turn back immediately. While the rest, most were fine and few (including me) were not able to exert much effort, but were otherwise also fine.
Note that this shows the safety margin is not all that big. For some people, 14,000 ft is dangerous, though for most it is not.
During WWII, US Army was supplying China via airlift over Himalayas and the route had minimum safe altitude of 19,000 ft. They had some strange accidents where the aircraft failed to round out after apparently normal approach and crashed at the threshold that happened mainly on the Chinese side where airfields were around 7,000 ft (while they were almost sea level on the Indian side). Investigation showed that apparently some pilots didn't use oxygen. Obviously they didn't feel unwell, but the reduction in mental performance showed. Strict order to wear the masks did resolve the issue, proving the analysis .
: Tunner, Lt. Gen. William H. (1955). Over the Hump; unfortunately the link on WP is dead, so I am linking the WP citation in the hope somebody will manage to fix it.