For Workspace, you can set a timeout to automatically log you out after a certain time. I'm not sure if this feature exists for regular accounts though, however even if it did it will not accomplish what you're suggesting which is to only remember you until you restart your computer. If you want to do that, then opening in a private window is the correct solution.

There also exists extensions and apps that can delete session data automatically, and Chrome has policies you can specify to only remember cookies for specific sites.