I don't get it
You say the chain wasn't that loose,then say with hindsight it was a bit slack.
Regardless of correct way to check tension or not,this guy wasn't checking the tension,he was checking to see whether it would ride off with the wheel turning,which if it did,it's either absolutely worn out,or far too slack. The difficulty is defining just how much pressure did it take him to get it to ride over the sprocket.
Technically though,if a chain can ride off the side of a sprocket easily,with very little side pressure,or with so little pressure even raising the bike on it's side stand would cause it(and yes,I've seen a few),the chain and/or sprockets are scrap. As for the M.O.T being two month ago,it's irrelevant.
I've tested bikes one week which can only just meet the minimum legal requirement,which means I can't fail them if they meet that,but a week later,sometimes less,they wouldn't be classed as roadworthy.
An example being a slightly dodgy chain and a rear tyre at just over 1mm tread. The guy goes out,does a burn out,screws the back tyre up and then gets pulled that afternoon.
In your case,I wouldn't know as I haven't seen the bike,and wasn't there,but if your chain was adjusted anywhere near correctly and in good condition,it shouldn't ride off easily. As I say,trying to define what lateral pressure he used.is the tricky part