Disable relationship path -> Disable all system relationship

Disabling connections path does not disable connections. You should still be able to request a connection perfectly fine from the menu above the users profile. Our demo site does not use connection paths tab as it's completely unpublished. All you should need to do is unpublish "Connection Paths" in CB > Tab Management. Leave "Enable Connections" and "Display Connection Path" as "Yes" in CB > Configuration > Connections.

Following your guidance, unpublishing the Connection Paths tab still makes the Request Connection tab also disappear.

These are my Connection settings in Configuration > Connections:

The request connection button in the banner above the users profile is in the connection path. There is no option to keep the button there, but hide the pathing. We've no plans for such an option as it'd just be a giant, mostly empty, banner with a single button.

When not using the connection path tab you request a connection from the menu above their profile. Please see the below.