I only know about a few states so the laws can be different in other states. In the handful of states I am familiar with, that is how it works. POA, durable or not, does not survive death. After death, decisions about the estate can only be made by the executor of the estate (at least if the asset is a probate asset) or the trustee if an asset is held in the name of a trust. Some assets in some states are exempt from probate, but it varies by state.
Frequently the person who is given durable POA is also named executor in the will. But if not, then the probate court has to appoint an executor. My guess is that they would be very inclined to name the person who held POA during life as the executor, but idk b/c everyone I know uses revocable trusts as will substitutes. You name yourself as the trustee of your revocable trust during your life and then you name a successor trustee should you become incapacitated or die. Any assets not held in the name of the trust are made payable/transfer on death. For those assets POD/TOD, you give someone durable POA should you b/c incapacitated.
That way you avoid the costs of probate on death. You never want to have to have your loved ones go through probate if you have assets. It is costly and it delays them getting access to your assets. But once you die, the person who had POA during life has no authority anymore. After death, the executor of the estate under the supervision of the probate court has authority to dispose of the probate assets, the trustee has it for any trust assets and the bank, brokerage, etc. where the assets that are POD/TOD are held disburses those assets according to the POD/TOD designation upon being presented with a death certificate.
Typically, the smartest thing to do is to put all of your assets in a revocable trust and/or make them payable on death/transfer on death in order to avoid the costs of probate. There are some exceptions to that, for instance, a primary residence. In states where they recognize a tenancy in the entirety b/t a husband and wife it is generally better to own your home as a TIE rather than put it in the name of a trust. Also, you may not get property tax breaks on your primary residence if it is held in the name of a trust. Joint tenancy with right of survivorship is another way to hold real estate that avoids probate but there could be issues with creditors of one of the joint owners if you hold title that way.
But again, all of that depends on the state you live in and if you are married and live in one of the few community property states, it gets really complicated.
And of course, if you are very wealthy there are more complicated vehicles you can use - like irrevocable trusts - but idk much about those. I just know for most of us who are middle to upper middle class, you are doing your loved ones a big favor by using a revocable trust, TIE, JTw/S and POD/TOD accounts instead of a will to dispose of your assets after death. In community property states it is more complicated. But at least in the states I am familiar with, POA doesn't survive death.