Gets rid of the largest set of mutable global state within the core.
This also paves a way for eliminating usages of GetInstance() on the
System class as a follow-up.
Note that no behavioral changes have been made, and this simply extracts
the functionality into a class. This also has the benefit of making
dependencies on the core timing functionality explicit within the
relevant interfaces.
These classes are non-trivial and are definitely going to be changed in
the future, so we default these to prevent issues with forward
declarations, and to keep the compiler from inlining tear-down code.
