I called airport immigration to enquire about the proper method. Seems like there is no clear policy. I spoke to four different officers, both at Don Meuang and Suvarnabhumi. The responses I got were:
At Don Meuang: No need to have the Amphur document, you can write a letter yourself, signed by both parents and attach your child's birth certificate.
At S airport: Three different officers: Get the the Amphur document. They have a form you can fill in.
As to the length of validity of the Amphur document, all of them said that a date 10 months ago would be too old.
So, to avoid doubt, wife and I trundled off to the Amphur. It took an hour. They only required the ID card of wife and child. (None of the marriage certificate, house registration, birth certificate previously required.) The whole process has been simplified and standardized.
As to whether or not you need the Amphur document, my guess is that it's not necessary and that you could probably wing it with a home made letter of consent. However, then you are subject to the discretion of the immigration official. With the official document there is no discretion.