To upgrade (not just patch) your kernel to the newest version livepatching is not sufficient, and you need to reboot. Note that livepatches include only high and critical kernel vulnerabilities. The service provides protection from serious security issues, but this is just one aspect of keeping your system secure and well maintained. There are a number of other software components that can require you to reboot your system, including, but not limited to:
CPU firmware and microcode updates
Updates to shared libraries and low-level dependencies (glibc, for example)
System BIOS and EFI updates
Enabling the Livepatch service does not turn on automatic installation of security updates in APT.
For best security, you should enable security updates using APT, subscribe to the security announcement mailing list, follow all advised security updates, and reboot your system at your earliest convenience when any software component requires it.
Additionally, kernel SRU updates include non-security bugfixes and lower-priority security fixes that may be important to your specific circumstances, and these fixes are only available by rebooting into an updated kernel.