Yes! If it wasn't legitimate, then there wouldn't have been five competitions for it. The Holy Grail takes roughly sixty years to regenerate after its use, slowly building mana over time and filling its cup. Once it's near completion, seven mages summon familiars that reflect heroes from across all ages to compete for a chance to win the wish. The magical energy used in summoning and giving form to those familiars then fills the rest of the Grail once they're eliminated. That incredible mass of power is what gives life to the wish.

[And the way the Holy Grail works is exactly what lead her to making assumptions about how the House works. She's quite sure that it feeds off the magical energy getting swapped around throughout, since said energy settles in the fluids connected to life (saliva and hair oils, blood, arousal fluids).]

Which is why I have no reason to doubt the wish at the bottom of are decks is real. It's getting to the bottom that's the challenge.