As both our DFs died and were buried long before our DCs were born, our DCs have been taken to both cemetaries since infancy. DS1 was about 4 when he made the connection that DH and I have DMs but no DFs and we have been open about their deaths and what death is.
A cemetary should be treated respectfully and is not a playground, but it's nice to see people of all ages using them to walk in, to sit in. I have a very pleasant memory of eating my lunch there with DM as we met up on my lunchbreak when I worked next-door.
Although thinking about it, our church certainly has no issue about it. At Easter, they set up an egg hunt through the graves in the churchyard and use it as a space for organised games for children's clubs.
I've never heard the sound of someone turning in their grave at the presence of children 