How I see you can implement this is:
https://example.com – This is the landing page/site where you have a menu on the top left and a user can choose “Private” or “Business”. You can set a cookie based on what user clicked on.
https://example.com/private/ – This could be an inner page which has it’s own template and loads a menu for private site OR this could be set up as a WordPress installation in a /private subdirectory
https://example.com/business/ – This could be an inner page which has it’s own template and loads a menu for business site OR this could be set up as a WordPress installation in a /business subdirectory
Next time when user lands on https://example.com your landing page template shall read the cookie and redirect the user to appropriate site.
I hope this helps. WordPress multisite is useful if you need to offer similar WordPress websites to a large number of users. So check your use case and keep multisite as last option.