Right, but if you can't sing to save your life it may be uncomfortable to attempt to do so.
Also Bards don't have to sing, and can do other things which the person may still not be able to do.
At this point most people choose something other than the bard. Letting the PC know it's okay to simply say "he pulls out his flute and plays a triumphant tune" instead of having to try to play a flute and sounding horrible is perfectly okay.
If you expect more from your players you are going to probably have to assume the bard will be passed over until you get someone truly talented wanting to play one.