Why is my Kenmore Dryer starting?

Me and my family have owned our Kenmore Elite HE3 dryer for decades and this is the first time that this has happened.

When you push the start button, it beeps while the “Check lint tray” light blinks. All of the settings are correct. My brother and I have dismantled it to clean it and we did it up spotless however it’s still beeping and blinking and not starting.

We’ve checked the (relatively) new heating coil and tried to reset it with the breaker method but nothing works. We’re starting to get frustrated yet not enough to call a real handy man.

Answers or suggestions appreciated.

@jayeff Thanks for the reply. The Model #: 11082822101