Why would the size of the final binary be so much smaller than the size of the static library?
Because you are not using all the functions of your library. A static library of archive type .a is a collection of .o object files and only the object files needed in your program are included at link time.
Whenever you statically link an executable, the linker can go ahead and resolve all symbol names (i.e. map them to an address) since all the symbols it will know about you have provided to the linker now (in the form of .o files and .a libraries which are really just a collection of .o files). If there are names that aren't there, you'll get a link error (this is different than dynamic linking where you may be able to load another library at runtime). In your case, you have extra symbols that are unreferenced by the executable. Since these symbols are known to the linker as being unused they are simply removed from the executable output. So your executable will be smaller than the input libraries in this case.