What are the theoretical reasons for treating the preposition as the head of a prepositional phrase? (Noun as head of NP sounds fine intuitively, but the same does not apply to prepositions in prepositional phrases. I have never seen a theoretical justification for this treatment.)
The head of a phrase ought to affect the category of that phrase. In turn, we can estimate whether phrases are of different categories by examining facts of verb subcategorization. Paradigms like
He fell into the hole *He fell the hole.
suggest that "into the hole" and "the hole" are of different categories, which will be true if the preposition "into" is the head of the complement of "fell", but not necessarily otherwise.
Heads are useful for understanding deep structure, but they can be hard to identify in surface structure.
Heads are closed-class words -- they are difficult to coin.
Heads have default values, which can often be elided if doing so will not confuse the delineation of other phrases.
Languages are pretty consistent about whether heads append to the beginning or end of phrases.
That said, nouns would not be the heads of phrases. The head of an NP would be a quantifier, but it can be omitted if the noun is proper, or the quantity is '[all]', or the quantity has been mentioned (definite). That is why some theories use DP instead of NP (and determiners don't have to also be pronouns).
Temporal nouns used adverbially don't need a head (preposition) because it can be assumed ('Sunday' = '[at] Sunday' = 'on Sunday') or it has already been grafted onto the noun ('today').
Verbs are headed by auxiliary verbs, unless those are omitted for the default (habitual/present tense or irregular verbs) or have been morphed onto the verb (regular past tense).