Something like this is done already in some liquid engines, but with propellant, not water, and not spinning (I don't think you can reliably spin anything inside a rocket engine, is too chaotic of an environment), but injected through the chamber walls.
In regenerative cooling, you circulate propellant literally through the inside of the engine walls, in both the cone and the chamber, and in some cases you can do something called film cooling, where the propellant is re-injected into the chamber or throat, or really wherever you need it, through holes in the walls.
Everyday astronaut has a very clear explanation in this article.
Everyday astronaut engine cooling rundown
Now I don't really think you can pull this off with water, it will likely boil off way before it even reaches the chamber (even ignoring regenerative cooling), you need cryogenic propellants, and you also need constant injection of new coolant because it will be quickly washed off by the combustion process.
The reason this is not always done even though it is probably cheaper from an operations and refurbishing perspective is simply because it is a more complex system, and the more pieces an engine has the more things can go wrong, the more expensive is its design, and likely requires longer design cycles. So at the end is a trade-off.
