if you want to make a dremel barbeque, you're on the right way!
jokes apart, your solution is very stiff and this will increase the finishing of surfaces, but it won't provide enough air circulation to avoid burning of rotor coils and brushes. Take in consideration that even the bearings are not he best, they tend to damage quickly if you apply relevant forces both axially or radially. My suggestion are two:
- buy a router milling unit from Milwaukee, Atlas Copco, AEG, Dewalt, Bosch... they cost about 100 - 150 $ with about 800 W of power, 10k to 30k rpm and spindle accomodating bits from 2 to 8 mm
- make a spindle by using a shaft mounted on two bearings, on one side mount the bit and on the other side mount a flexible coupling to be attached to an electric motor. The intermediate solution would be coupling Dremel with flexible shaft to your homemade spindle, but take care because as i wrote above Dremel is intended for small and quick works, not for milling continuosly for a day.