At the core, there is no difference between Shutdown or Reboot, with regards to the "System Restart Required" message. Both a shutdown and a reboot will clear it. However, this only applies when you don't have a new pending update that requires a reboot to completely apply, and automatic updates could run on your system since the 'last reboot' you mentioned.
As such, you need to be mindful of whether your system has unattended-upgrades
installed and enabled. If this is the case, your system gets updates once a day or so, and if you have automatic updates set up to install all available updates (not just security-only updates), then it will autorun and autoinstall updates at its configured time point. The best way to determine that is to look at /var/log/apt/history.log
, where automatic updates will show up. This can explain an 'unexpected' "Restart Required" message because since the last reboot your system might have gotten new updates that triggered the message.