(Note: I am at a late-beginner / low-intermediate learner of French, so I welcome others to point out inaccurate information that I might write in this answer.)
Would it be wrong to say: "Il est seulement parti depuis trois jours,
mais le me manque terriblement"?
It would be wrong to say that, for the same reason that it would be wrong to write the sentence alone: "Le me manque terriblement". The correct sentence is instead "Il me manque terriblement".
The word il is a subject pronoun. Common subject pronouns you've probably learned about are: je, tu, il, elle, nous, vous, ils, elles. (You might have also learned about on).
Most sentences (that is, aside from greetings like "Bonjour") need a subject! "Le me manque terriblement" does not have a subject. ("Le me manque terriblement" literally translates into "Him is-missing to me terribly", which doesn't work in English, either; but "He is-missing to me terribly" does work, because it has a subject (the subject is "he"))
If you were to say I punch him it would it not be je le frappe? Why is
this different for I miss him?
There are at least two important things that you need to understand, in order to translate "I miss him":
- Understanding direct vs indirect objects; and then using object pronouns to replace a direct object or an indirect object.
- The surprising (for English speakers) way that manquer à works. I assume you already understand this from your readings, but hopefully the examples later on in this answer will further clarify your confusion.
Direct vs Indirect Objects
Consider the following two sentences:
- I phone Marc. Je telephone à Marc.
- I punch Marc. Je frappe Marc.
In French, indirect objects always begin1 with a preposition (almost always à or de4, but sometimes pour 2), and direct objects never have a preposition.
In "Je téléphone à Marc", "à Marc" is an indirect object.
In "Je frappe Marc", "Marc" is a direct object.
You can use an indirect object pronoun to replace an indirect object3 :
I phone Marc. Je téléphone à Marc.
--> I phone him. Je lui téléphone.
You can use a direct object pronoun to replace a direct object:
I punch Marc. Je frappe Marc.
--> I punch him. Je le frappe.
Here is another example. My own name is David; suppose I say these sentences:
Alex phones David. Alex téléphone à David.
--> Alex phones me. Alex me téléphone.
Alex punches David. Alex frappe David.
--> Alex punches me. Alex me frappe.
- you see that in "Alex me téléphone", me is an indirect object pronoun, because it replaced an indirect object ("à David").
- you see that in "Alex me frappe", me is a direct object pronoun, because it replaced a direct object ("David").
Your question title says: "Direct object pronoun with Manquer à". Now you know that when using "manquer à quelqu'un" ("to be-missing to someone"), you won't ever use a direct object pronoun; you will have to use an indirect object pronoun, because when you see à quelqu'un, you know that it is an indirect object.
Now suppose I speak these sentences (remembering that my name is David) :
Il manque à David. ("He is-missing to David", or in more natural English, "David misses him").
--> Il me manque. ("He is-missing to me", or in more natural English, "I miss him")
Hopefully you can now see why "Il me manque terriblement" is correct, and why "Le me manque terriblement" is incorrect.
Possibly helpful reading (though I didn't read these carefully, so I'm not sure helpful they actually will be) :
Footnote 1: Some French grammar websites/textbooks will say (as I do here) that "à Marc" is an indirect object; and some will say that only "Marc" is an indirect object, but that it is preceded by "à". In practice, both views work; there isn't any real consequence about how to analyse grammar (e.g. you won't be deciding on which indirect pronoun to use any differently, etc, regardless of which view you take)
Footnote 2: Indirect objects might be introduced by prepositions other than à, de, and pour -- I don't have 100% understanding of this. But it seems that, when only considering indirect objects that can be replaced using object pronouns, à, de, and occasionally pour will always introduce the indirect object.
Footnote 3: Choosing the correct indirect object pronoun to replace an indirect object isn't straightforward; I'm choosing not to try to explain how to choose the correct indirect object pronoun in this answer.
Footnote 4: Some French grammar websites and textbooks consider objects introduced by "de" to not be indirect objects; and some French grammar websites and textbooks do. In either case, "de + noun" is replaced with the pronoun en. The French grammar websites that consider "de + noun" to not be indirect objects will talk about replacing them with the "adverbial pronoun en" instead of replacing them with an "indirect object pronoun en"
The textbook "Advanced French Grammar" by Monique L'Huillier definitely calls "à Marc" (ie, including the preposition) an indirect object:
and both this textbook, and the textbook "The structure of Modern Standard French" by Mosegaard Hansen, seem to consider objects introduced by de to be indirect objects; whereas all the websites I linked to only consider objects introduced by à and pour to be indirect objects.