My theories below may still be valid, but I finally found a direct answer in that other comment
https://www.duolingo.com/comment/24304566 (Luchtmens/Adrian P. would know)
The short pronouns are required for accusative persons (îl văd pe tatăl), and they are used optionally but customarily in order to sound less formal with dative persons ([îi] dau tatălui).
Then I am even more confused. I just had a "choose all the valid sentences" question at the end of this lesson that counted me off for only choosing a translation of the sentence "The boys read to the men" that included the "le" that doubled "bărbaților" and not also choosing a sentence that omitted the "le."
No, verbs are transitive and require accusative, but apparently nouns that represent persons are not accepted as accusative objects, hence the workaround with "pe" and the need for the accusative pronoun.
But this is only my quirky reasoning, I'm no grammar scholar. It just occured to me, that direct objects of transitive verbs wouldn't accept prepositions like "pe" (Latin "per"?). So, if "văd Maria" won't do for some reason, "văd pe Maria" leaves the transitive verb incomplete, requiring "o văd pe Maria". Spanish has a similar issue, but I forgot if they need a pronoun in addition to "a Maria"...
My personal theory is that for transitive verbs "pe ..." is not a proper direct (accusative) object, just a prepositional complement, like when I say "I see in all your glory" - the direct object "you" is missing, and "see" would become intransitive, as in "yeah, I see."
Dative, on the other hand is not even necessary to make the verb complete: "I give a puppy" doesn't need the dative boy as a receiver to make the transitive verb (I give sth.) grammarwise complete - but we need the accusative puppy!
Maybe Romanians use the short dative pronouns only because they are used to them from accusative sentences, even when they are not really necessary?
Here it is dative (acestui = to this) and not genetive (acestui = of this), but in no way it is nominative or accusative (acest = this). With acestui as demonstrative pronoun, băiat doesn't need an article (-ului) anymore, but is still dative (just happens to look the same as nom/acc băiat without the article).
So, just to clarify, if this is the first language with cases you have looked at, each noun, adjective, and pronoun has several forms. "Acesta" can be either the nominative form (the subject of the verb) or the accusative form (the direct object of the verb). "Acestui" is either the genitive form (generally, OF the thing, often used in possessive, as "its") or the dative form (TO the thing, often the indirect object of a verb). A Romanian word, apparently, can also have a separate vocative form, which would indicate that the thing was being directly spoken to, e.g. "How are you, Jake?" German has even fewer cases, but it you are interested in Czech or Russian or, God help you, Hungarian, Turkish, or Finnish, you will have to deal with even more such cases, as these forms are called. The whole system of the cases is called declension.