Can an appointed POA represent named executor of a will on their behalf?

Usually you can apply to the court to do so, especially if there is no other family ready, willing and able (wanting?) to serve. But you cannot do this automatically. You have to go to court with your papers and with your willingness and ask the judge if you can serve as administrator of the estate for your husband while he is disabled and unable. If there is no contest you almost certainly will be given letters testamentary to do this tough duty. You will be signing things. It is important you have the legal ability and papers to do so. Good luck.

It is up to the judge. If the named executor is not able, someone else will be appointed. If the POA applies and isn’t objectionable for some reason like too young, a felon etc and no one else objects or applies then the judge can appoint whomever appears to be the best candidate. Look up the rules for executor in California. Rules can be state specific.
I am both my aunts POA and her named executor.

If he's competent, he can carry out his duties as executor. If not, I'd consult a probate attorney. I don't see a POA being able to involve themselves at all.