By this reasoning, isn't it okay to kill someone (or at least to kill them in their sleep)?
You tell me.
If you don't like the moral implications of a certain hypothesis, this should have precisely zero effect on your estimation of the probability that this hypothesis is correct. The entire history of the growing acceptance of evolution as a "true" theory follows precisely this course. Many people HATED the implication that man is just another animal. That a sentiment for morality evolved because groups in which that sentiment existed were able to out-compete groups in which that sentiment was weaker. That the statue of David or the theory of General Relativity, or the love you feel for your mother or your dog arise as a consequence, ultimately, of mindless random variations producing populations from which some do better than others and pass down the variations they have to the next generation.
So if the implications of the continuity of consciousness are morally distasteful to you, do not make the mistake of thinking that makes them any less likely to be true. A study of science and scientific progress should cure you of this very human tendency.
If your reasoning implies ~X, then X implies that your reasoning is wrong. And if X implies that your reasoning is wrong, then evidence for X is evidence against your reasoning.
In other words, you have no idea what you are talking about. The fact that something has "distasteful implications" (that is, that it implies ~X, and there is evidence for X) does mean it is less likely to be true.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.