Although not about your problem, this thread has a lot of useful information relevant to your situation.
https://www.smokingmeatforums.com/threads/mes-rewire-simple-guide-no-back-removal-needed.267069/
These units work by switching the neutral side of the element to the incoming line. When it is off, you will find 120V to ground on both sides of the element. When the element is heating, the neutral side will be 0V (or close to it.)
In decreasing order of probability:
1) Your overtemp switch has failed. This is the round disk you see on the back wall of your smoker, sometimes called the "rollout switch" because the same part is used in gas furnaces for that purpose. The switch disconnects the neutral wire if the temperature gets too high, and reconnects when the temperature falls. I believe newer units have a small access panel on the back side that can be removed, likely rivets drilled out, to facilitate changing it. Otherwise, you can remove the entire back panel by drilling rivets, or cut a smaller access panel as outlined in the thread I listed above.
2) The relay has failed. The relay switches the neutral wire to control the temperature in the smoker. The relay is a part of the relay and power supply board located in the compartment on the bottom of the smoker. It is accessed by drilling the rivets out of the access cover located there. Replacement boards may or may not be available for your particular unit.
3) The wiring connections to the overtemp switch have corroded and no longer make contact. This is possible, but based on how your connectors to the element look, it seems unlikely as they typically fail first.
4) The controller has failed. Even though you can set the time and temperature, it still has to be able to drive the relay. I would think this would be a rare occurrence.
So, what to do next....
I would drill the rivets on the bottom panel to gain access to the relay board. I would then prepare a short 14 or 12 gauge wire with male .250 quick disconnects on each end. Referring to the thread linked above, use the jumper to connect the two wires going to the relay, #2 and #3 in the photo. #1 and #4 are already connected by the board. The element and overtemp switch are are now connected directly to the power plug. Next, plug it in and see if the element starts warming up. If it doesn't warm up, the problem is the overtemp switch or the connections to it. If it does warm up, it is the relay or the controller. In this case, I would make sure the jumper is well insulated with electrical tape, screw or pop-rivet the cover back on and go buy a PID controller as you have already rewired your smoker to use one.