score:6
According to wikipedia, Winston Churchill had a state funeral which was held on Saturday, January 30th 1965, so the answer is "no", state funerals are not always held on a Monday.
Upvote:3
This may be relevant to the thinking behind the question.
It was planned well in advance that the funeral of Her late Majesty, our dearly beloved Queen, would be on the tenth day after her death, unless she died on a Thursday in which case it would be the 11th day after her death. As it happened she died on a Thursday so her funeral is Monday.
The reason for Thursday being 11 is to avoid having a funeral on a Sunday, the Christian holy day.
So there was no rule that it had to be a Monday. It could have been any day but Sunday.