I think we limit the rights and privileges we extend to animals in the same way that we arbitrate competing interests in society. To that end, we grant rights to others in society except in those cases where, a) doing so critically interferes materially with a more primary right or need of other individuals, b) is at odds with a compelling interest of society, and c) when extending that right would lead us as a society to behaving in ways which would endanger legitimate and compelling societal interests. (Empathy is an example of the latter; independent of animal rights, we don't want to encourage sadistic behavior towards animals by children, as that would lead to sadistic behavior towards humans.) I'm basically a strict speciesist and extend rights and privileges only inasmuch as it furthers our interests as a species, analogous to the situation between individuals and a society: the species is the society, and non-humans get rights only when doing so is in the interest of our species or essentially of zero cost. I reject the slippery slope, though there are obviously going to be cases where there is sufficient doubt where the gentle part of the slope extends for us to grant similar rights and privileges to an animal that in some ill-defined sense can be one of us, though they may not strictly be a part of our species.
This is a quick off the cuff reply, so I haven't considered it in any real depth, beyond my basic speciesist stance. In particular, I haven't analyzed how well it does or should describe how we deal with marginal cases (e.g. children, the mentally handicapped, people in a persistent vegetative state, coma patients), nor does it likely adequately grapple with questions of vagueness, those cases that arise because of the indistinctness of boundaries (your neanderthal hybrid), nor does it speak to other prominent hypotheticals like machine sentience. Maybe I should have had my coffee first.
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