Does a power of attorney automatically become executor of an estate,
if the deceased in question has no written will?
No. Even in states where that person may have priority to be appointed as executor, this requires a separate application to be appointed and is not automatic.
At common law, authority under a power of attorney as a power of attorney agent terminated at the moment of death automatically by operation of law. Under a common statutory reform of that rule adopted in Colorado and some other states, however, a power of attorney agent's authority is not terminated until the agent learns of the power of attorney principal's death.
In U.S. states that have adopted the estate administration provisions of the Uniform Probate Code (not many), the priorities to be appointed to administer the estate as a personal representative (a.k.a. executor) are as follows (using the numbering of Colorado Revised Statues § 15-12-203(1), which is one such state):
Whether the proceedings are formal or informal, persons who are not
disqualified have priority for appointment in the following order:
(a) The person with priority as determined by a probated will
including a person nominated by a power conferred in a will;
(b) The surviving spouse of the decedent who is a devisee of the
(b.3) The surviving party to a civil union entered into in accordance
with article 15 of title 14, C.R.S., who is a devisee of the decedent;
(b.5) A person given priority to be a personal representative in a
designated beneficiary agreement made pursuant to article 22 of this
(c) Other devisees of the decedent;
(d) The surviving spouse of the decedent;
(d.5) The surviving party to a civil union entered into in accordance
with article 15 of title 14, C.R.S.;
(e) Other heirs of the decedent;
(f) Forty-five days after the death of the decedent, any creditor.
Under that statute, a holder of a power of attorney does not have any priority by virtue of having been a power of attorney agent.
A civil union is a (usually) same sex relationship equivalent to marriage, a status created before same sex marriage was legalized. A "designated beneficiary" is someone given the inheritance and incapacity related rights of a surviving spouse to someone who isn't a spouse or civil union member, to the extent that a more specific document does not provide otherwise.
Many other U.S. states have a similar list of priorities under this question governed by state law.
This said, someone who does have priority to be appointed as personal representative or executor, and is subsequently appointed, can ratify actions taken before appointment if they are in the best interests of the probate estate, and has limited authority to stabilize the property of the estate (e.g. making sure that doors to buildings are secured or that cattle are fed) pending appointment not longer thereafter.