Old heater burnt through the "coating" From the terminal to the burnt through area, it had continuity. This portion was also touching the inside of the metal heater cylinder.

I replaced the heater, but I do not get 240V between the two heater terminals. I get 240V at the power coming into the first main relay, and individual 120V off of each relay leg leading to the heater when heat is activated.

What would cause me to not see 240V across the element terminals?

The contactor (relay) is most likely there to complete the circuit only when it's ok to give you heat. If there is not enough flow, (pressure Switch) If the high limit is triped (high limit switch), then there will be no power to the coil. (not to be confused with line/load terminals). If your coil is not reading voltage (usually 110) then the logic is doing it's job, and you will not bridge the 220 volts. If the coil Does have it's proper voltage, and does not bridge, then the contactor is bad. Seeing 110v on each load terminal also suggests a failed contactor only supplying one leg of the 220v, and mirroring the same leg back to the other pole of load side via the heater element. To sum it up,
Coil voltage present means if ya got it here (line 220 ) then you should get it there ( load 220 )