You can't send the money without the private key corresponding to that output. Nothing about including an output in a ring signature gives you access to the private key. The whole point of ring signatures is that you can construct a ring signature using only the public, not private keys of the other possible signers.
You have only your own private key, so you can only spend your own outputs.
It guarantees that someone in the group has the private key that enabled him to sign a transaction spending the output. This allows an observer to verify that the output has been spent by the authorized party (someone with the private key) but does not allow the observer to determine which of the group is the authorized party.