Should I be concerned with the accumulation of road cinder in my garage?
I dont think there is much to worry about.  Living in the snow belt in Ohio, our garage is almost completely black by the end of the winter.  We try to brush or even spray it our a few times if the weather permits.  But that is really to keep it from tracking into the house, not so much for the floor.

I would be more concerned with salt on your garage floor than the cinders.  The salt can certainly eat away and break down the concrete.

There are concrete sealers that you could apply in warmer weather than might help to protect it, but I have never heard of anyone doing that in a garage.