I think I can now see why we can't agree on this.
Take temperature. People will commonly say that all the temperature scales are relative, except for Kelvins, which are called absolute. Now if you apply your arguments for enthalpy to temperature, you will say that K is relative as well- relative to the 0K which we have to define as temperature at which only zero point internal energy is present in subtances.
Now, when I say that enthalpy (as a state function, not reaction enthalpy changes) is absolute, I mean exactly the same type of absoluteness as in the temperature case above. Realistically relative to a defined zero state.
Obviously, in this way everything is relative. Having two apples is relative to having zero apples etc. It's just a question of defining the "ground" level.
The only difference of course is that we can't directly measure enthalpy easily (but I guess we can work it out from calculating internal energy from the quantum solutions + macroscopic energy?), but we can easily measure temperature. Hence the need to use enthalpy changes.
That's all good - my argument that enthalpies are always zero or positive stands and is as good an explanation for the original question as using enthalpies of formation.