That’s because all assets transferred to the trust are no longer owned by the person establishing the trust (here, the decedent). And, with an irrevocable trust, the value of the trust is not considered as part of the estate for estate tax purposes.
If all of the father’s assets were appropriately transferred to the trust, there is no estate to probate and the executor is superfluous. If there was a pour over will, any remaining assets would be transferred to the trust. The trust likely named a trustee, who will then administer and/or distribute trust assets to beneficiaries as per the trust document.
Also, some states require that the will specify if and what the executor may be paid for doing that job, otherwise it’s simply expense reimbursement. With a trust, the document should specify how and if the trustee will be paid.