If when you simply add skype command to the Startup applications, Skype does start-up, but there is no icon in the system tray, so that if you close the main window, or have the option to start in system tray, you cannot access Skype anymore with a GUI. Even worse, the Skype instance continues to run on the background, preventing you from opening a new one.
To me, the solution was instead of simply adding a command skype, to add
bash -c "sleep 10 && skype"
This waits for a bit before starting up Skype, so it seems that the problem was that Skype was starting up too early before some other necessary things were ready.
Source: this ask Ubunutu page