Operation of infinite switch
Very revised answer:

Very annoying.
 While my original answer to the original question was largely correct - ie the actual load in the element has no effect on timing, it was substantially wong in another area. I'll go back and edit everything in due course, but for now, original answer is left intact below.

I originally could not find a diagram of the internal connections of an "infinite switch" anywhere. Based on what is written elsewhere it sounded as if the heating coil connected across AC mains. However, this diagram which Dave found on this very interesting fire investigation site tells a different story.

More detail below, but a surprising conclusion is that
 Doubling the element wattage will, for a given setting. 


reduce power by about 30% at high heat settings,  
reduce power output by 70% at low settings.  
reduce power output by 50% at medium settings. 


You asked about affect of load (eg pot) on element power.
Notionally pot does not load element BUT I'd expect for a given power that element would get hotter due to loss of radiation path and some convection losses and pot reflection.
 If element is eg Nichrome this will not effect it. It will NOT have NTC so if it is mildly PTC (positive temperature coefficient of resistance) then res will rise with covering. Power in element = V^2/R so power will go down which will TEND to increase power level from the infinite switch as above. This should be a small effect and depends on Relement being PTC.

The above lower ower wity more Watts element effect suggest manufacturers may want to make switches to suit element powers.

The electronic circuit below dos not depend o an element in series with the element, unlike the mechanical one. 



Background:

The switch's heater element is IN SERIES with the range element.When the heater is cold the switch is on and the range element will be turned on. The heating energy will be governed by the element current which will depend on the element' Wattage. So doubling the wattage doubles the current.
 HOWEVER the series heater will only dissipate a small percentage of the main voltage and the voltage drop across the heater element will be proportional to current if the heater resistance remains constant. As a consequence heater watts
will rise with the square of the current. So doubling element wattage increases heater power by 4 times. If on time is approximately proportional to heater power then doubling element wattage will reduce on time by a factor of 4. Off time can be expected to be close to constant as temperature at turnoff will presumably be similar and cooling time should be largely unrelated to heating rate. So, imagine that a controller was running at 66 time units on, 34 time units off. Doubling element wattage can be expected to reduce on time to about 16 time units. So duty cycle goes from 66/99 = 2/3 = 0.666 to 16/5= = 0.32 or 2:1. That sounds very good.
 BUT if originally running at say 20:80 = 0.20 it would change to 5/85 = 0.06 or 30% of original.
 At the other extreme, of running at 90:10 = 0.90 it would change to 70 % of original.
 ie Doubling the element wattage will
 reduce power by about 30% at high heat settings,
 reduce power output by 70% at low settings.
 reduce power output by 50% at medium settings. 

This means that a higher Wattage element will reduce output for a given setting and have an increasing rate of getting hot a the top end.



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No - the temperature of the heating element or an item being heated has no effect on these controls.  Here in New Zealand one name they are called is a  "simmerstat" or an "energy regulator"- they will probably have other names elsewhere. 

They are what is known as "open loop" controls. Only the dial setting controls the on/off ratio - the load or target object is NOT "sensed". As the dial is turned the on/off or mark-space ratio changes. 

Controllers do exist that take account of the target temperature. This is usually termed a thermostat if you are heating air. These also may use a bimetallic strip but it is controlled by the target air temperature.



Here is an elecronic simmerstat control schematic. Note that they could sense load power but have chosen not to. OM1895 IC - diagram from here.  

10 page application  note