I apologise to the answer

Question

Their head sounds like a myth, with an eye for why
Endless, it says to meet but briefly so,
standing for the very famous MA institute.

Then comes the first name of another puzzler, all too familiar
Named who is after the prime of a time wheel with
a Norse god at his feet, starting six places away.

At last, anger or annoy without end—
its birth half the death and equally four sided
of any trio’s total member number.

I apologise to that person. But who are they?

Bonus: Find out why.

Answer 1

It's Mithrandir

His head sounds like a myth, with an eye for why
Endless, it says to meet but briefly so,
standing for the very famous MA institute.

Mith is the head, sounds like Myth, also i (eye) -> y (why)
Supposedly Mit is pronounced similarly to Meet (Thanks Sid)
MA (Massachusetts) Institute -> MIT, front of Mithrandir

Then comes the first name of another puzzler, we all are very familiar with
named after the prime of a time wheel with
a horse god at his feet, whose kind really starts six places later.

Next is Rand.
First name of Rand al'Thor, someone we are familiar with because they're #1 in Rep here.
Named after the book series, Wheel of Time.
Assuming horse -> Norse, Rand al'Thor, is 6 places into their name. (Apparently wrong)
We're supposed to take the h of horse and add 6 to get an n for Norse, to make the Thor. (I prefer the first version)

At last, anger or annoy without end—
its birth is half the death, where the dawn is equally four sided of
any trio’s total member number.

Ir is the end.
Ire - e = Ir || Irk - k = Ir, endless anger/annoy.
I (9) is half of R (18). And is 3^2 -- one day, from Rubio.

Bonus:

There was a dispute about you posting their rhyme.