I think that the obvious example is a fine one. If you can do it in the obvious case, then you obviously can do it. Also, the obvious case you mention answers the OP's question since it would set the potential to 0 at the surface where the field is non-zero.I tried to think of a practical example of applying GPE where it could actually help to use anything other than Infinity - except for the obvious ones where we are starting off from earth and falling back again.
Also, you wouldn't necessarily use the mgh approximation, e.g. in launching a satellite or ICBM.
I agree. Gravitons are completely irrelevant to this discussion.That could be why I come out in spots at the mention of gravitons - except when the person who introduces them is a BRAIN.