Local would be not using register flags at all and instead indicating with each operation whether you want flushing or not (and rounding mode, ideally). Some libms may clear and restore the control flags and some may not. Libm is just an example here and one where you're right that most of function calls that might need to avoid flushing subnormals to zero are expensive enough that clearing and restoring flags is an acceptable cost. However, that's not always the case—sometimes the operation in question is a few instructions and it may get inlined into some other code. It might be possible to handle this better at the compiler level while still using the MXCSR register, but if it is, LLVM certainly can't currently do that well.

In theory, every function should do that to check things like rounding mode etc. But that would be pretty slow, especially for low-latency operations (modifying mxcsr will disrupt pipelining for example).

That wouldn't be practical. C math library performance really matters for numerical-intensive apps like games.