[lldb] Make thread safety the responsibility of the log handlers

Drop the thread-safe flag and make the locking strategy the
responsibility of the individual log handler.

Previously we got away with a non-thread safe mode because we were using
unbuffered streams that rely on the underlying syscalls/OS for
synchronization. With the introduction of log handlers, we can have
arbitrary logic involved in writing out the logs. With this patch the
log handlers can pick the most appropriate locking strategy for their
particular implementation.

Differential revision: https://reviews.llvm.org/D127922