Letting your child "make friends" at kiddie arcades/kids places?

Okay so today we went to this place that was a cross between a Chuckie cheese/Dave and Buster's type place with all the usual arcade games, but also a kids ball zone/ jungle gym, bumper cars, and Lazer tag. Well we were there with my in laws and my sister in law had her four children, I had my almost 3 year old son and my baby girl.

Anyways, the kids are playing, their mom left to go to the bathroom, leaving me to watch her two youngest in the ball zone (5 and 3 years old.) Her 8 year old ran up to me with another little girl and asked where her mother is. I told her she had gone to the bathroom and would be right back and to just wait for her. Well they both had game cards and played a game near me for a moment until I had to get up with my two month old to tell my sister in laws son that he could not climb up the slide because he was going to get hit by a kid coming down. This other little girl convinced my SILs daughter it was fine to sneak off into the Lazer tag game instead of waiting to ask. No one could find my SILs child for over 20 minutes or this other little girl.

We had to involve the employees to look for her since our search part of six adults and one 12 year old could not find her anywhere. Naturally we feared the worst. They finally found her and this other little girl, who's mother hadn't even thought to look for her daughter now 30 minutes after they'd disappeared, and the other mother acted like it was fine that they had disappeared and didn't seem concerned that her daughter had disappeared with another little girl. This little girl also came back later and tried to get my niece to go with her a second time and told her to "run and find her" when no one was looking.

After this I'm wary of letting my kids play with other kids at places like this. My niece could have easily been lured away by another child and snatched. Now luckily that wasnt the case, but it was what we feared when we couldn't find her. Shes been taught not to go with any strangers, but kids naturally trust other children.

Would you let your child run around with another kid at a children's arcade or a similar place? After this I don't think that I could. They won't make any lasting friends after an hour at a play place like this, and you never know what another kid can convince your hold to do, even if your child "knows better" than to do something. They'll do it to win over another kid they want to keep playing with. These kids were supervised all except for the two minutes that eyes aren't directly on them when they ran off.