I've never noticed it not starting after the install, but then I don't do the client installs, the platorm teams at my site do them - and they usually reboot a client after the install anyway.
I do know that the VSP kernel driver, which is installed by default on Windows clients, requires a reboot before it can be used by NetBackup. But, if you don't want to use VSP then you can turn that off (i.e. disable the install of VSP) by modifying a flag in the silentclient.cmd script. This way you don't need to reboot before using the NetBackup client - and you can use VSS instead of VSP, if you want to at least try to backup open files.
So, maybe it's the default install, i.e. the act of installing the VSP kernel driver that causes the client service not to be started after the install. I'm not sure. Maybe someone could do some testing and confirm actual behaviour based on different installation methods/types.