As a concrete example: in Arc-F, it's possible to overload '+ by simply overloading <base>+. If you have a new type foo and want to define how to add two foos together, you just overload <base>+. The neat thing is that this won't slow down arithmetic very much, if at all; if you were to write the equivalent overload '+ in Anarki (Anarki doesn't have the <base>+ hook, after all), arithmetic performance suddenly drops precipitously.
Also, overloading <base>+ is simpler: <base>+ only requires that you consider the case of adding two objects. + determines how to handle things when you do (+ foo1 foo2 foo3 foo4), specifically it converts the call to (<base>+ (<base>+ (<base>+ foo1 foo2) foo3) foo4)
So yes: my position is, by focusing on efficiency in implementation, it frees us to actually use the axiomatic approach without throwing in the towel and going non-axiomatic.