So the motor is being stressed - gets hot and quite probably a thermal, safety-overload switch is operating.
The motor will be getting stressed and hot because...
System bearings tight
Mixing bucket overloaded
Insufficient cooling
Defective motor / motor brushes
I am not an expert in mixers - I just guess at a distance.
Quite why the shop people suggested a new switch and lead I don't know - they were obviously guessing too.
Good luck...
Peccavi
May 2010