I agree that the copies will diverge almost instantly; I'm just saying that small amounts of divergence aren't a big deal. That's what I'm trying to illustrate with my example of the person who loses an hour of memories. I think this is exactly equivalent to making a copy, having that copy exist for an hour, and then destroying it. An hour of memories does make the copy different from the original, but the loss of the copy is just the loss of that hour, not of a complete human being (and we naturally quickly forget much more than that - I already can't remember what I did every hour yesterday).
I admit I don't feel like it's exactly equivalent, but I think that's an illusion caused by my moral intuitions developing in a wold where destroying a copy always means destroying the only copy.