Does Irish allow the concept of an implied direct object when using a transitive verb? For example, in English one could write “my cousin wrote to me…” where the direct object “a letter” or “e-mail” is implied.

I’d say in this case, “a letter” or “an e-mail” is implied in the speaker’s mind, but it’s not implied grammatically.

We were talking about ullmhú, but I think another verb with a similar problem is “nigh”.
I guess you can say “I washed” in English (to say “I washed myself”). English is full of verbs like that, that don’t require a reflexive pronoun even though the action is being done on yourself.
However, in Irish I don’t think you could say anything like “nigh mé” instead of “nigh mé mé féin”. Nor “caithfidh mé ní” instead of “caithfidh mé mé féin a ní”.