I verified -with the dryer running - that I'm getting 240 V at the heating element (120V on each terminal),
If you checked the heating element for voltage referenced to Neutral/ground and saw 120 volts on each leg and then assumed that this was the same as 240 volts across the element then you checked it incorrectly.
If the element is good and you see 120 on one side you will also see the same 120 (phase) on the other side. Checking the heater voltage using Neutral is confusing as you do not know what phase side you are reading.

You must check across the element for 240 volts.
If when running and there is 240 across the heater then it must heat up unless there is a heat coil problem.

Try flipping the breaker off/on slowly a couple times. Sometimes you can loose half the line without actually tripping the breaker.
Check the voltage at the plug
L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
If OK
Unplug the unit and check the wires at the terminal strip in the machine to make sure none are loose or burned out
If OK
Check the power at the terminal strip.
Do this with the heater off and on.
[COLOR="Red"]Be careful as 240 volts is lethal !!![/COLOR]