Here, how about this: our 'rights' are not granted by the Constitution, they are emphasized, publicly proclaimed. The Constitution delineates the rights and responsibilities of the GOVERNMENT. Anything granted can be taken away.
For: Let's enumerate some rights, just in case future generations are stupidly literal about things and get confused because it's not all written down for them.
Against: Bad idea! By writing some down, we might create the impression that those are the ONLY rights.
For: Well then we'll make the 10th Amendment a catch-all, so it's crystal clear that rights exist beyond those we've enumerated so far.
------
So did the 10th Amendment work? It seems like probably not, since people so frequently refer to the Bill of Rights as proof that the Constitution "grants" rights--rather than the intended effect, which is that the Bill of Rights limits government power over rights that already (and always) exist.