I guess there are several ways you can go about it. You're right that Spurs don't necessarily need to keep Manu's rights, but it all depends on how much money (or lack thereof) he's willing to take. Once you renounce his rights and start using that cap space, then your only methods for re-signing him are:
a.) cap space
b.) "the room exception," which is basically an MLE available to teams who were previously under the cap; I think that value is a max contract of 2 years, $5 million, but I don't know if that changes for this year, or
c.) he could sign for the minimum
So, if you assume that Ginobili wants more than 2 years/$5 million, then you either have to sign him using Bird rights, or make darn sure you have enough cap space to re-sign if you renounce those rights.