Using coolant as 3rd propellant

Question

Rocket engines often have 2 propelants, one of which is used to cool the engine and after that the hot propellant is injected in the combustion chamber.

Would make sense to have a third propellant more suitable for the cooling process?

As an example, at high temperatures ammonia decomposes endothermically into H2 and N2 which can then be injected into the combustion chamber, but likely injected at a much lower temperature due to the endothermic process, ammonia have a much lower energy density than methane or hypergolic fuels but the decomposed products have similar energy densities.

Answer 1

Early liquid fuel rockets kinda did this, by adding water to the fuel (V2: 75% ethanol, 25% water).

Another related example is the Viking engine, which uses water to cool combustion gases entering the pump. The water does however not end up in the combustion chamber as the engine uses a gas generator cycle.

But once you have reliable regenerative coolant cycles, it makes little sense adding a third propellant. A "better coolant" would ...require less coolant? But when the coolant is the fuel itself, there are no mass savings in using less coolant, as long as there's enough of it.
And there's typically enough of it. Only a fraction of the fuel is redirected to cool the nozzle, and expander cycle engines are even limited by waste heat!

Another side to it is that when looking for coolants that are at the same time reasonable rocket fuels, it turns out that most rocket fuels are already good coolants.