There could be a temperature sensor that is at fault (those are very very cheap) or it could have something to do with the element or the timer. Will the dryer timer advance on a timed cycle vs an auto dry cycle?

Okay, the timer is working just fine then. The way those Amana/Speed Queen dryers work for the auto dry cycle is by thermostatic control. The machine starts off the cycle by heating until the temperature sensor is satisfied and then it begins to advance the timer. If the machine is not heating, it won't advance the timer in the auto dry cycles as the criteria for advancement of the timer are not being met. The timed dry cycle ignores the temperature level and advances for a set number of minutes no matter what the temperature in the dryer reaches.

I would check the thermostats for continuity as well as the element. It's possible that the venting is clogged with lint. When that happens, the dryer can overheat and that will trip the overheat thermostat which stops the element from being able to heat. Checking and cleaning the lint filter is not enough, the vent lines from the inside of the dryer to the exterior vent port on the outside of your house should be checked at least once a year and cleaned out. It could also be due to wax build up on the lint filter due to extended use of dryer sheets or fabric softener. Wash the lint filter with dish soap and a soft brush. In either case, the thermostats would still need to be replaced. The heating element will be on the more expensive side whereas the thermostats are very cheap.

I'm not saying that cleaning the venting and lint filter will fix the problem. If they are clogged and they've tripped the thermostats, the thermostats will still need to be replaced. However, if that's why the thermostats were tripped, they'll need to be cleaned in order to prevent the new thermostats from tripping right away causing the same problem.

There are two kinds of thermostats at work here:

The first is for normal cycling on and off of the element to maintain temperature in the drum, it's possible that there is something wrong with this one causing it to never turn the element on.

The second is an overheat thermostat, this one is like a fuse, if they dryer gets too hot then there is likely an issue with clogging or some other problem that makes the dryer not safe to run with the element. If this one is tripped, the element will not be allowed to come on at all until the thermostat or "fuse" if you will, is replaced. The reason they do this is to force one to open up the dryer to look for other issues that caused the overheat to prevent fires.

They are usually located right next to each other. It's very possible the temp regulating one is just fine and the overload thermostat is blown. This happened to my dad's dryer recently when my sister didn't change the lint trap for 5 consecutive loads.

The general location is usually somewhere on the blower housing.

Hope this helps!