And if I say 'I have this one thing that cures my hayfever reliably, and no other thing does, but it has no mechanism except for the placebo effect', is that very different from 'I have this magic thing?'.

No, not very. Also, if it turns out that only this one thing works, and no other thing works, then (correcting for the usual expectation effects) that is relatively strong evidence that something more than the placebo effect is going on. Conversely, if it is the placebo effect, I would expect that a variety of substances could replace the sugar pills without changing the effect much.

Another way of putting this is, if I believe that the placebo effect is curing my hayfever, that ultimately means the power to cure my hayfever resides inside my brain and the question is how to arrange things so that that power gets applied properly. If I believe that this pill cures my hayfever (whether via "the placebo effect" or via "magic" or via "science" or whatever other dimly understood label I tack onto the process), that means the power resides outside my brain and the question is how to secure a steady supply of the pill.

Those two conditions seem pretty different to me.