If you schedule a restart as an /updateCenter "task" (appears in the list of plugins to be installed, see screenshot), and then cancel the quiet down from the build queue on any page (instead of unchecking the checkbox), /updateCenter's checkbox and restart functionality breaks until the next Jenkins restart. This broken state survives a 'reload configuration'.

The attached screenshot on a new test instance I set up for this shows what is shown after performing these steps:
0. Start a job that runs a while
1. Update a plugin
2. Schedule restart using checkbox
3. Navigate away
4. Cancel restart from build queue
(now the build from 0. may finish)
5. Update another plugin

The restart never happens, in fact, all executors are idle and the queue is empty at the time I took the screenshot. The checkbox cannot be unchecked, it always "re-checks" itself now.

This can easily happen on an instance with long running jobs: While waiting for the 10 hour build to finish, you might cancel the restart temporarily to let a few other quick builds run. Unfortunately, you won't be able to resume the restart in Update Center.

Without logging in to the server to restart the service, or having e.g. Safe Restart Plugin installed (provides a management link), you need to know about the /safeRestart URL and type it in restart Jenkins after this occurred.