My instinct would be that food will cook quicker just before the water starts boiling because steam conducts heat slower than water and so when the food is in contact with the steam bubbles it receives less thermal energy.
This is absolutely the opposite of reality!
A steam bubble in contact with a body with a temperature less than
100°C will condense on this body, transferring about 540 cal/g!
In fact heating a wall by condensing steam is one of the most
efficient heat transfer methods.
But now your primary question: those noodles "cook" faster
where they experience the higher temperature. In a pot on a fire,
the surface of the water will be as low as maybe 80 or 90 degrees,
when bubbles raise at the bottom, but do not reach the surface yet.
(Again here: heat transfer by bubble evaporation is extremely
fast, same reasons as for condensation)
10 or 20 °C less are a lot when cooking noodles (wheat starch)
this may lower the reaction rate to half the value at 100°C,
or even lower (rule of thumb for such reaction).
Thermal conductivity: most amorphous solids/liquids have similar, low
heat conduction. Crystalline solids are medium, metals are much higher,
the thermal conductivity being related to the electrical conductivity.
Google for: Wiedemann Franz Lorenz.
Thermal conductivity of gases can be calculated using kinetic gas theory,
this was "triumph" for Clausius, Maxwell and Boltzmann.
In the real world, you are approximately right and the cooking time is pretty much independent of the number of eggs. An egg only cares about the temperature of the water that surrounds it and as long as it is kept near the boiling point at all times, an egg can't possibly "know" how many siblings it has. So its gradual change (and its rate) is universal, independent of the number of eggs.
On the other hand, if this is some textbook example about proportionality, it's one of the would-be real-world applications of proportionality and the underlying assumption is that the cooking time is proportional to the number of eggs. In the real world, this can only be true if the number of eggs affects the actual temperature of the water around them.
A justification of the proportionality is that the egg cooker needs to pump a certain amount of energy $E$ to an egg to turn it into a hard-boiled egg. Because the egg cooker consumes the same amount of energy per unit time, it needs $N$ times longer time to pump the energy $NE$ into the $E$ eggs. In practice, this may be a somewhat reasonable description if the egg cooker is weak enough and it has a hard time to achieve or maintain the boiling point as the eggs are cooling the water down.
The truth in the real world will be somewhere in between: the cooking time may grow with the number of eggs but the growth is much slower than the direct proportionality.
When I am cooking various deeply frozen lunches, the recommended cooking times for half a bag are often something like 80% of the cooking times for the whole bag. The idea is that with more food on the pan, it takes a bit longer for the heat to penetrate from the pan to the interior of the food. The idea with the eggs is similar if the cooking time is supposed to increase with the number of eggs.
Best Answer
Presumably, the rate of the steam escaping the cooker depends on the "resistance" of the steam path: from the opening in the bottom, where the steam enters the dome, to the opening on the side of the dome, from where the steam escapes.
The more eggs in the cooker, the narrower the path, the slower the flow. Also, as relatively slowly moving steam makes contact with more eggs, it is more likely to condense and make its way back to the water at the bottom of the cooker, which further reduces its escape rate.
So, with more eggs in the cooker, a smaller amount of water will last about as long as a greater amount of water with fewer eggs, resulting in a similar degree of cooking.