because the paradox states that the barber shaves all the men’s heads.

You are missing the fact that both of the statements in this riddle are excludent. If someone else shaves him, he would become a part o the group he shaves (men in the village who do not shave themselves).

Consider two groups: A – “men who do not shave themselves” x B – “men who do shave themselves”.
The barber could never fit in any of those groups, because being in group B would make him a part of A and vice versa.

If the barber has a long beard it literally means he doesn’t shave himself, so he would become a man that he shaves. The same logic of this paradox applies to all paradoxes. If there was a solution it wouldn’t be a paradox.

Thanks for your question! No one else can shave the barber, because the barber himself must shave all the men in the village who do not shave themselves. So if the barber is a man who does not shave himself, he must be shaved by the barber. But since he is the barber, that would mean that he is shaving himself, which creates a problem, because the barber must shave only the men who do not shave themselves. This problem cannot be resolved by having someone else shave the barber, because then the barber would be a man who does not shave himself, bringing him back into the category of men who must be shaved by the barber.