It is worth stating plainly what the three way match is for, because the case for the upstream discipline it needs is usually made in terms of efficiency when it is really about loss prevention. Three specific losses are prevented by comparing the invoice against the order and the receipt, and no other routine check in a payables process covers them.
Paying for what never arrived
The goods receipt is the only routine evidence that a delivery actually happened. Without it, an invoice for goods that were never delivered looks exactly like an invoice for goods that were. This is the loss the third document exists to prevent, and it is the reason two-way matching is a genuine reduction in protection for physical goods.
Paying a price nobody agreed
The purchase order records what was agreed, at the time it was agreed. Comparing the invoice against it catches price increases applied without notice and pricing errors, both of which are common and neither of which anybody notices from the invoice alone. Tolerance decides how large a difference passes unexamined, which is why setting it deliberately matters.
Paying twice for one delivery
Matching at line level with running quantities means a second invoice against an already-satisfied line does not clear. Document-level matching misses this, which is one more reason line level is worth insisting on. Combined with duplicate detection on supplier and number, it closes most of the double-payment paths.
Questions people ask about 3 way match accounts payable
Is the match worth the upstream effort?
That is exactly the question, and it should be answered in terms of these three losses rather than efficiency alone.
What if we cannot get receipts recorded?
You are running a two-way match with the delivery risk unmitigated. Worth saying explicitly, so it is a decision rather than a drift.
Does automation change the control?
It performs it consistently rather than when somebody has time, which is a real improvement. It does not change what the control is.