If they have to send your Doggie down for the long nap, bring a big blanket he likes to the vet. Sit on the blanket, put his head on your lap, and they'll administer the meds to him right there. Just rub his neck or whatever until he conks out for the last time (down here at least).
It happens very, very fast and there is no drama. I'd say a kid of over 12 or 13 would probably be okay with the whole situation (being there, petting the dog or cat, etc.) but I'd 2x check with the DVM and of course you have kids and I do not.

But in my book, that's the 'right way' to do it. When you send them off to wherever they are going, you should be the last person with them.