You guys brought a couple of interesting points and the conversation does veer into Chaiya Boxing as you will see.
Monks in Thailand at one time taught a spectrum of things at the wat(or temple) including religion, martial arts, sometimes medicine, etc and could be looked at as local schools scattered throughout the kingdom.
" In the past the Wat was the centre of community, particularly of education. The Wat belonged to everyone in the community. The term buad rian clearly indicates the relationship between wat and people in terms of education......Nevertheless, for all intents and purposes, the role of monks as educators has long ceased. "
You could consider this early period "premodern Thailand" with modernization beginning under King Chulalongkorn in the late 1800s into early 1900, where things began to radically change throughout the kingdom. In the late 1920s, Old style bare knuckle considered brutal and barbaric was banned and the new sport Muay Thai rose up fusing westernized rules of sportsmanship, gloves, timed rounds, and a structure of rules.
Monks and their role in society was more greatly defined and monks now concentrated on teaching religion and nothing more.
When we talk of Muay Chaiya, a monk brought this form of bare knuckle fighting to Chaiya and spread it from there. It rose to be a formidable fighting style and produced many of the top bare knuckle fighters of the era. So the very existence of Muay Chaiya is credited to a monk teaching and spreading it from the South of Thailand.
One of the younger students of the late Kru Tong who lived with him after his own father died, became a monk. He felt that it was in the tradition of Muay Chaiya that it was fair for a monk to teach it in private since that was how it began. I trained with this monk for a long time and he taught everyone for free and people in the community supported him though many disagreed and looked down upon what he was doing as wrong especially teaching women. He recently disrobed and returned to worldly life. But he taught for many, many years in different parts of Thailand.
I think there is a certain romantic notion attached to all of this. People automatically say its better and/or superior and that is not always the case. The monks are focused and have tremendous amount of energy. Sometimes, monks have no choice but to defend themselves as in the South of Thailand, where Muslim terrorists attempt to burn down temples or kill the monks after dark in Muslim heavy areas. I met monks in the south very skilled in weapons and muay. In Burma, many of the monks i knew and trained with practiced martial arts and many died in the sad government crackdown against them and now they are heavily monitored since the protests initiated by the monkhood against the government.
But if you are specifically addressing Thailand then you can understand why you see so few monks devoting some part of their teaching to fighting either muay thai or any other form of thai martial arts. If they do, they will generally do it in private as not to draw attention to themselves but as a personal extension of their own teachings.