Cutting too slow can produce more heat. If you broke the shaft of the router bit and you were burning it then I suspect that you were trying to cut too deep per pass.That job goes much easier if you do the rough cut out with a jig saw, band saw, or coping saw.
You have another problem still and that is grain direction. If you look at the factory made steering wheel you will notice that there is no end grain showing. Using a solid board like that you will be crossing end grain on 2 sides and the end grain sections will be very prone to break. The most reliable way to make that would be to take straight grain and overlap pieces to make a loop, kind of like the sections of a wooden barrel but overlapping sections so that all the joints are reinforced.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.