Except for kernel modules - which are stored on disk and often loaded on demand. Though most distros keep around the old kernel and kernel modules to avoid issues. You can still get issues on distro such as Arch which replace the kernel and modules on an upgrade - though there is the kernel-modules-hook package with pacman hooks to keep the old modules around until the next reboot to fix this.

If you don't have the kernel-modules-hook package installed on Arch you really should reboot shortly after an upgrade to the kernel or else new devices you plug in might not function until you do (typically USB as those tend to be what you hotplug the most).

there is live kernel patching, most distros (like Arch Linux) however don't set this up by default
there is also the possibility that one live patch don't work properly
the only reliable way currently to load the full new kernel is via reboot

the real way for server however is to not rely on one single server but have a redundant amount which you can reboot one at a time

You don't need to change kernel if the one you have works fine.

For gaming PCs you're often running hardware that gets improved in the kernel often. For servers, as long as you're not trying to squeeze every once of performance out of it you can get by with way behind bleeding edge kernels.

One of the easiest attack vectors to secure on a kernel is compiling all your modules directly into the kernel and disabling loadable kernel modules.

Once you've got a kernel that has all the bells and whistles you need compiled into it, it's relatively mature enough to have it's bugs worked out (like an LTS kernel that's been out for a month or so), and you've applied proper system and kernel hardening configs, that server can run undisturbed for quite a while.