I reckon it'd work OK – water is a perfectly valid choice for a hydraulic fluid – it's certainly been used in the past, and in places where leaking oil is unacceptable people still use it. Problem is that hydraulic fluids serve other functions beside simple power transmission, so:
-It may dissolve more air than oils, which could come out of solution and cause problems when it gets warm
-It has a low viscosity, so leakage might be an issue (but seals on brakes are pretty good, so probably fine)
-It won't lubricate as well as oil – you might notice the extra friction/stiction. However, having played around with dry hydraulic brakes, they're pretty smooth, so I doubt this would be an issue
-It has a lower boiling point than oil – this is the main issue (but not for trials riders….). Over time, car brake fluid absorbs water, dropping the boiling point to a level that eventually gets dangerous. New DOT4 boils at 230C.
I do wonder though whether you'd mostly get away with it – the thermal conductivity is probably rather higher as is the heat capacity, so the performance is possibly closer to oil than you might expect…. The dissolved air might be the real issue though.
Note also, that cooking oils have few of these limitations – they're in fact designed for high temperature operation!
Personally though, I might stick with DOT 4 and mineral oil 😉