Very old topic, but by far the most common burnt smell from a vacuum is the belt.
Typically, thread, hair or string builds up around the roller bearings, and when they finally tighten up enough to slow the belt, the motor shaft starts to melt through the rubber belt.
I used lots of vacuums in my cleaning business. All of them were free. All of them either needed a new belt or the bag changed. All worked well regardless of cosmetic condition.
Only once have I run across bad motor brushes, and in that instance it caused intermittent operation, when the worn brushes made momentary contact.
It seems that people, using their vacuum only a few times a month, think that after a few years, when it just "doesn't have suction" (full bag) or it "smells funny" (burning belt from seized roller) , their vacuum is old and needs replacement.
It is rare that vacuums get enough miles on them to really kill the motor. Usually, some essential plastic tab on the body breaks that makes the vacuum no longer worth dealing with.