I carefully cut the gristle away from the meat, and discard it. I sometimes throw away the fat, too, but sometimes eat it. It’s bad for me, but it’s delicious.
I’ve never heard of being forced to eat the gristle. If you are so poor your family needs to eat it, you’re likely hungry enough to eat it without being prompted. I don’t think most Americans eat it, though. I mean, maybe a little that’s hard to remove, but not big chunks.