init: Combine the CheckShutdown() and set_do_shutdown() methods

Let the CheckShutdown() method clear the do_shutdown_ member instead of
clearing that member separately from calling CheckShutdown().

Bug: 266255006
Change-Id: Ifc1cff2be92a45db7f91be2fdb812930d2fd1ad5
Signed-off-by: Bart Van Assche <bvanassche@google.com>