Thinking on this a bit more, the reason I think taking-but-not-giving is anti-social is because society only works if people give. If everyone does their bit, society is just a little bit better for everyone.
But what if organ donation isn't about society at all? No give and take. It's just a completely isolated, magical gift, with nothing expected in return. I can see some people would wish to look at it that way, and would want to actively get a donor card, with nothing expected in return, to make this magical gift possible.
I can see that an 'opt out' system would take the shine off this somewhat for them. But if they then opt out, are they not putting their own feelings about whether something feels 'special enough' for them potentially above someone else's life? Is that right?