Keep in mind that if you're checking for power at the element with wires connected, your 120v at both sides of element is likely a reading through the element itself. What I mean is that you're reading 120v where it supplies element, then again after it passes through the element, then through your meter to neutral or ground.
It's quite possible that the 120v you're reading is, by design, supposed to be powered all the time, with the other side of the 240v circuit being switched by the thermostat / safeties.
Point being, that just because you're reading 120v doesn't mean that the stoves' controls are calling for the element to be energized.
I do commercial / industrial HVAC work for a living, and usually get around this stuff pretty well, but every time I end up working on household appliances.....once I start following the shematic.....I'm amazed at how complicated they make these things.