Actually it's not an Elive issue in the sense that you're running E24 and "connman" as opposed to the default E16 and "network-manager".
I agree though and see the same issues you describe so clearly there.
Solution: Install "connman-gtk" and run that.
The "internet" application you show there is the frontend of "network-manager" and running that in concurrence with "connman" will give issues.
So chose for either one (E24 - connman centric) and disable the other:
"sudo systemctl disable --now network-manager.service" or with "connman.service"
We've actually been through this before: 
