Ubercart override default flat rate quote
What I discovered is that in order for the default shipping flat rate quote to be overridden by product flat rate quote:

In Configuration->shipping quotes:
Base price must be set to 0.00
Default product shipping rate set to whatever the flat rate is.  In my example, it was $7.00.

In product ->shipping settings->flat shipping rates:
Flat rate quotes should be set to the overriding amount.  In my example, it was $3.00.

So, in the test product example, the flat quote shipping rate would be $3.00.  If no rate is set in the product, then the rate will be the default shipping quote of $7.00.

A bit confusing at first, but logical, I guess.