> "Regalian functions of the state" is a common term
It's really not; I mean, I've got a poli sci degree so I've seen it, but it's not really a common term.
> that means the basic powers of the state (monarchy or not) that used to be the prerogative of the sovereign. I.e. law & order, military, and taxation.
The pre-modern-limited-governnent prerogatives of sovereigns were much broader than law enforcement, defense, and taxation. And, in fact, the principle thing denoted by regalian power is the fundamentally ownership of all land which is superior to all private title, which is the root of all the other (essentially unlimited) powers associated with sovereignty, in a “my house, my rules” kind of way.
The term “regalian functions”, though, is most often used to refer to the three functions Adam Smith identified as essential to the sovereign in The Wealth of Nations, which are still much broader than what you suggest:
“The first duty of the sovereign, that of protecting the society from the violence and invasion of other independent societies, can be performed only by means of a military force.” [0]
“The second duty of the sovereign, that of protecting, as far as possible, every member of the society from the injustice or oppression of every other member of it, or the duty of establishing an exact administration of justice, requires two very different degrees of expense in the different periods of society.” [1]
“The third and last duty of the sovereign or commonwealth, is that of erecting and maintaining those public institutions and those public works, which though they may be in the highest degree advantageous to a great society, are, however, of such a nature, that the profit could never repay the expense to any individual, or small number of individuals; and which it, therefore, cannot be expected that any individual, or small number of individuals, should erect or maintain.” [2]
> Hence military is definetly in a different category than healthcare.
This is true, not in the sense of not being subsumed within “regalian functions”, but in the sense that the military is the first regalian function identified by Smith while healthcare delivery systems that broadly and effectively serve the whole population are in the third.
[0] book V, chap. 1, part 1
[1] Ibid, part 2
[2] Ibid, part 3