How to evaluate the tragedy of death
the tragedy of a
person's death for the person who has died ought to be evaluated in
terms of the goods that the person had prudential reason to care about
at the time of death. The weaker the degree of psychological unity
between the person at the time of death and the person at the time the
good in question would have been realized in her life, the less
important that good is for evaluating the extent to which the person's
death was a tragedy for her. (Rahul Kumar, 'Permissible Killing and the Irrelevance of Being Human', The Journal of Ethics, Vol. 12, No. 1 (2008), pp. 57-80 : 60 - summarising Jeff McMahan. The Ethics of Killing: Problems at the Margins of Life (Oxford: Oxford University Press, 2002.)
McMahon calls this the Time-Relative Interest account of the tragedy of death. So far as it goes, I see nothing objectionable in it. But let's see how it can be put to use in the context of the present question. It may look different then.
Degrees of moral wrong
His (McMahon's) initial suggestion
is that what is fundamentally wrong with killing - here he has in mind
the killing of all kinds of animals, including human beings - is that it
frustrates the time-relative interests in continuing to live of the victim.
To a certain extent, this fits with certain intuitions about killing. The
suggestion that the killing of animals is not morally objectionable is
morally perverse, but it is plausible to think that that the killing of a
person is a much more serious wrong than the killing of an animal,
just as the killing of a mouse is less seriously wrong than killing a
dolphin or a chimp. This is so for two reasons, both having to do with
the psychological capacities of most non-human animals. First, non-
human animals do not have the same range of goods available to
them; some goods require complex reasoning and planning abilities
that non-human animals do not (to the best of our knowledge)
possess. Second, the psychological capacities of non-human animals
do not allow their lives to be as psychologically unified as the lives of
humans can, in principle, be; certain goods, such as living a life that
has a certain character, requires the possibility of that kind of unity. (Rahul Kumar, 'Permissible Killing and the Irrelevance of Being Human', The Journal of Ethics, Vol. 12, No. 1 (2008), pp. 57-80 : 61.)
I shall term this the Kumar-McMahon view. A claim here worries me : 'it is plausible to think that that the killing of a person is a much more serious wrong than the killing of an animal, just as the killing of a mouse is less seriously wrong than killing a dolphin or a chimp.'
I'm not at all sure that I would accept without qualification that the killing of a mouse is less seriously wrong than killing a dolphin or a chimp. It would be wrong, equally morally wrong, to kill any sentient being merely for fun or to pass the time - mouse or chimp.
That aside, the claim that 'the killing of a person is a much more serious wrong than the killing of an animal', even if we assume that neither killing is done maliciously and both are done to flourishing specimens, does nothing to show that it is not morally wrong to kill an animal but only (at most) that 'the killing of a person is a much more serious wrong than the killing of an animal'. It in fact concedes the point that the killing of an animal is morally wrong.
Re-evaluating the tragedy of death
Quoted above : 'the tragedy of a person's death for the person who has died ought to be evaluated in terms of the goods that the person had prudential reason to care about at the time of death.' There's an important phrase here :
'the tragedy of a person's death for the person who has died'. It would be a tragedy if the last healthy white rhino died, even though the rhino had no prudential reason to care about certain goods at the time of death - indeed had no capacity to care about them.
My overall impression is that the Kumar-McMahon argument does not deny but endorses the moral wrongness of killing healthy animals for whatever purpose. Nor can I see how the 'range of goods' and the 'psychological unity' considerations justify using (killing) animals for our own ends despite the moral wrong we do them.
Complexity all round
The morality of our treatment of animals, particularly our mass-killing of them for food (let alone sport), is immensely complex. I have only set out, and criticised, one argument, an argument that rests on (the assumption of) our being an intellectually superior species. I can see no valid moral inference from 'We are an intellectually superior species to (other) animals' to 'It is morally right or permissible to kill intellectually inferior species, (other) animals, for our own purposes'. There's a missing premise : 'It is morally right or permissible for an intellectually superior species to kill intellectually inferior species, (other) animals, for its our own purposes'. But is it ? This is the very point at issue.