My 4, 6 and 8 yos are alone with my dog whenever I leave the room. I come and go, and so do they, and I don't even think about it. That has been the case for about two years.
I trust my dog, and I trust my kids. I've never had a problem with the dog or with the kids, who treat her with utmost respect.
I cannot imagine living where I had to remove the kids or the dog whenever I left the room. Our dog is a part of the family, she knows her boundaries, and the kids know how to treat her.
Of course, something could go wrong, just the same as the kitchen could catch fire or someone could break into the house. I see it as the same sort of 'risk', and I am not prepared to live my life in fear of what freak thing might happen.
Now, if I ever saw any sign in my dog to say that she couldn't be trusted, I'd rethink, but she has been trained well since a puppy, has good bite inhibition, and has never given me even the slightest, tiniest cause for concern. For me, the reason for having a dog is to have her incorporated into our lives. If she's had enough of the action of the household, she goes to her crate, and the kids know to leave her alone if she's in there. If they have friends around who I don't know well, I crate her. Other than that, she's around the kids and me, either playing with them, or doing her own thing. That's how it was for me with dogs growing up, and that's how I want my dog to fit in with our family life.