I have... really mixed feelings about this story.
To the left, as a teacher, I kind of resent stories that play up the whole "those silly teachers, they don't really know what's best for our children" angle. When it comes to education, I do know better than you because I went to freaking school for it for two years, not to mention six years of classroom experience, and I don't give a damn about your twelve years of parenting experience anyway because we have totally different goals, damn it! You might know your kid better, but I know how to teach.
Sorry. Hot-button topic.
So, anyway, it kind of annoyed me that the story's drama came from creating tension with a stupid, sort of straw-man-ish education policy. I don't think the author was actually trying to be critical of teachers or education - I think she just picked them as a plot element to create tension, and it happened to hit a nerve with me.
To the right, I thought this was a great example of classic sci-fi, exploring how mankind would react culturally to a rather reasonable extrapolation of real-life technology; in this case, the dramatic lengthening of life caused by safe and simple organ transplant technology. It was an interesting take on the age old "oh noes, the technologies!" phenomenon, with one generation insisting that a younger generation had to learn this valuable lesson... even though that lesson maybe isn't really important anymore, in a world where pets - and, the story seemed to imply, possibly eventually even people - don't have to die.
Overall, I thought the story was interesting. I found the great grandfather's position much more attractive, because I've always been in the "mankind strives to overcome all obstacles and rule the universe" camp. Death is dumb, and we should transcend it. The mother was an effective POV character, though I wasn't entirely clear what her position on this issue was supposed to be - perhaps that was the point.
What annoyed me most was the cat. I'm sure that Greycat's apathy was meant to provide a counterargument, that by not dying we would doom ourselves to fade away into apathy. The story presented it as a position, but didn't really back it up. Why was Greycat apathetic? What about being an immortal cat would do that to you? Why wouldn't he just keep on running around... catting? Great grandpa was at least as old in human years, and he was anything but apathetic.
So, overall, I would say that this was a very good story, though not a perfect one. The point seemed a little... murky. Not unclear because it's complicated, but unclear because the presentation was a little opaque. However, it was certainly fun and thought-provoking, so I can call it a success!