Root cause: the final restart block called Get-WebAppPoolState first to
decide Restart- vs Start-WebAppPool. With \Continue = 'Stop',
any transient error from that check (most likely right after
Wait-AppPoolFullyStopped had to force-kill a lingering worker process,
leaving the pool's WAS state briefly inconsistent) threw an uncaught
error and aborted the rest of the script - silently skipping the admin
site/pool start even though the public site had already restarted
successfully by that point.

Fix: Start-WebAppPool/Start-Website are no-ops if already started, so
drop the state check entirely. Both public and admin site/pool starts
now go through Start-SiteAndPool, which wraps each call independently
in try/catch so one failing doesn't prevent the other from starting,
and reports (rather than swallows or aborts on) any failure.