Whose call is it to place a declining spouse with dementia to be placed? The neurologist or the spouse?

You make the decision. No doctor has that power. The AL may just want something from the doctor asking what DHs healt problems are and his diagnosis. Actually, the AL will evaluate him for what type of care he needs.

And do not take your husband to see ALs with you. You go and pick out the one you think is best suited. Taking husband will cause problems. When we placed my Mom, I didn't tell her until we were in the car going. (She was in her last stages of Dementia) We told her she was going to a new apt and would be making new friends.

A doctor is not God as some of them think they are. My daughter has found the younger ones coming up seem to not have the God complex the older ones do. Also, they work better with the Nurses.

If you can not safely care for him at home then it is your call to place him in a facility that can care for him. that could be MC or SN or even AL if it is safe for him.
If the neurologist disagrees with you then the neurologist can take your husband home and care for him.
YOU are the caregiver, YOU are the POA, YOU are the one that makes decisions as to his care.

Make some appointments for you and your husband to go visit and see if has a preference. Getting him into care benefits BOTH of you, if he is willing to visit his future, the transition will be much smoother. Consulting the doctor and requesting referrals will create a "Team," but you are the captain of this ship unless he becomes belligerent then you'll need a Geriatric Psychiatrist.

Doctors are people too and each one has their own personal bias based on their belief system and life experiences. A doc can express the opinion that someone can still be managed at home, but that is based solely on their brief acquaintance of that person and does not take into account what care at home actually looks like, it can't because they don't know anything about that. Are you hoping that someone else will make this choice for you?

Cheeky, let me put it to you this way--the neurologist doesn't get to tell you what to do.

S/he can't say " your health and well-being don't matter. You must continue to care for your DH at home not matter what the financial or human cost."

You get to say when you're at your limit and make other arrangements for his care

The Spouse if you have POA. The hubby, if not competent to make good decisions in his own and your behalf doesn't count other than weigh in. You are doing the work. The MD can recommend and can suggest social services and what support you can get, but you are the one with the tough decision. We have seen caregivers go down before their loved ones needing 24/7 care. You must make this decision for yourself.

I’ve not heard of a doctor being involved in having a patient move anywhere beyond making a general recommendation. The family decides when a person moves to onsite care

It's the caregiver/spouse's call.

You ultimately have the final say(I'm assuming you're hubby's POA)as to when your husband is placed, but I guess it can't hurt if the neurologist is also on your side as when hubby asks why he's in a new place(if he does)and when he's coming home, you can say that his doctor said he needed to be there and can come home when the doctor says he can(wink wink).
I'm not sure what you're waiting for though as you've been burned out for quite some time. It's now time to act and do not only what is best for your husband but for you as well.
Best wishes in finding the best facility for him.