Can a POA that has become seriously ill appoint the executor of the estate to handle affairs while they get well?

POA and executor are two different thinks. A person appoints POA to handle their affairs if not able before their death. POA stops at death. An executor is appointed, usually, in a will. That person goes to Probate and makes sure all debts are paid and then makes sure those mentioned in the will get what was bequeathed to them. Two different responsibilities. They don't need to be one and the same.

A P.O.A. can stay on as 'overseer' and the only one with true authority but can arrange for someone else to do the hands on care - once P.O.A. is resigned it can't be reassumed - it would be good to get some type of 'succession' in order before the surgery

Then the only child will have to have her surgery and as soon as she is able resume the duties as POA. No one can just appoint another to assume responsibilities designated to the first party. A POA is not like a lease. It is a binding contract only with the person with whom the giver assigns. Notify all of mom's business creditors in advance or pay in advance of surgery.

What exactly is the POA wanting done that can't be delegated or put off for a few months? Bills can be paid online or set up with auto-pay, Mom's doctor's appointments can be rescheduled (although that really isn't a duty requiring a POA). Remember that POA is the authorization to make decisions for someone else - financial or medical depending on what kind you are talking about - it is not a commitment to hands on care.

POA has no authority to transfer any powers to anyone. Read the POA document to see if a standby POA is named there. If none is named, the only remaining option is to seek Guardian status.