Thanks again for answering (and to Fernando for suggesting the elementary argument). It's helpful to know where the well-pointed assumption is used.

I understand the argument using the Quillen adjunction when the spaces are q-cofibrant (in the pointed Quillen model structure), because then the left derived functor of the adjunction is given by smashing with A (on objects). What i dont understand is how this works under the assumption that X and Y are h-cofibrant, since then one has to q-cofibrantly replace before smashing.

What would the induction step look like, smashing source and target with a cellular pushout and then use some sort of gluing lemma ?

Ah, thank you. I did not consider this bijection because i don't know how to proof it. One might try to show that if p:C->X is a cofibrant approximation (of diagram spaces or just spaces depending on the setting), then p/\A:C/\A->X/\A is again a cofibrant approximation when X is well-pointed, but this seems like circular reasoning.