Whether or not you phase food out really depends.
If you weren't paid to go to work, would you still go? The treats, the tugtoy, the scent, the chase are all "payments". The value of the payments depends on the dog- some dogs will accept payment in treats, others just want to run their agility trial.
All of us have been told a lie, "the dog wants to please their master", when really, that's not the case. They do things 'to please their master', either because they get punished if they don't, or because they find it reinforcing in some way. Any way you slice it, they are still being R or P, depending on the situation.
When I take my dog out, I pay her for doing something she doesn't want to do- she would probably rather run all over the place, jump on people, play with dogs, get into things, etc. But I have molded the situation and the reinforcers in a way that she finds that it's a better deal to 'behave'.
I think I grow more and more as a trainer, as I kind of sink into and embrace this idea.