> The other approach is to expand what land is available to build on. I understand that in the UK, farmers consider it a lottery win if they get planning permission for some of their farmland to become housing.

I've seen that mentality in other parts of the world. The thing is that agricultural land is often protected because it is good agricultural land. It is literally the reason why the cities were built where they were built, since that land provided for the cities. Unfortunately, good agricultural land also tends to be easy to build on and we have a tendency to ignore features that may make it a poor decision to build on. (Such as the flood plains you mentioned.)