Yes and no.
He speaks through me
isn't really a natural sentence, but even so...
He is my spokesman
He represents me
is what you could interpret it as.
Remember in the above context
sé = he
mé = I (as in "me")
... which you have confused
What you suggested...
I am his spokesman
Labhraíonn mé tríd / Labhraím tríd
Sounds like something from the Bible.
Huh? I don't see why you think fgalla confused the pronouns... it seems to me that fgalla's interpretation makes more sense than yours. If "labhraíonn sé tríom" literally means "he speaks through me", then yes, it seems to me that could mean "I am his spokesman", just as fgalla said. "Labhraíonn mé tríd", if it means "I speak through him", would be "he is my spokesman". A spokesman speaks FOR a person, but a person speaks THROUGH a spokesman, not the other way around. Unless these prepositions work radically differently in Irish than in English, which is entirely possible, but in that case the issue comes from misinterpreting the prepositions, not confusing the pronouns.
Ceapeann go bhfuil... "he speaks through me" means that he speaks to the person on the other side of me hence ignoring me
That is how I have heard this used in english- Teachers in primary school would often say- Speak to him not through him- meaning address him not me (if the student was talking about the student to the teacher in the 3rd person as if the other student was not there)
When addressing a committee you would speak through the chair. When addressing a court you would speak through the bench. When addressing the House of Commons you would speak through the Speaker.
In another usage, when speaking though someone I indicate that he is speaking for me (i.e. on my behalf).
I have also heard this structure used to indicate that someone is essentially ignoring another person. Not so much drowning him out, but speaking to a third person as if the ignored person were not there.
And how do you deal with the squabbling husband and wife? "Tell your father his dinner's in the dog!"