The unit not running after stopping is probably the speed control board.
The 400 is easily taken apart like you said so you may as well do this again and check for a broken circuit from the plug up to the switch and to the speed control board. Don't forget to also check the common neutral as well. When you are satisfied the wiring from the plug to the on switch and past thru to the speed then you have your answer.
Next log into the Dremel web page and fill in a request for repair.
In the US if it is under a year they will fix it for nothing and even older will have you send it to a repair depot and you will not get charged then they will pay the return shipping.
You did not say where you lived but they also have online chat and the person can give you a repair number. Rockwell/Dremel if asked will usually repair for nothing you just have to wait for them to do it.
In Canada go to the Canadian web page and it is almost the same. I have had this on my 300 and 400 and have been used good each time. Tell them the whole story including you taking it apart.
Lots of luck
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