I'd say that in many places its difficult for 16 year old's to get a part time job.
Supermarkets and other companies that employ under 18's regularly will already have staff, and usually have a lot of 6th formers desperate for hours before uni.
As said seasonal employers have usually hired their staff by now.
The suggestion of "coming up with a game and booking a stall at every fete over the summer"? Lots of barriers to entry there too.
Car washing? You'd have to be competitively priced with the valet places that are all over big cities or the auto washes are relatively inexpensive.
Lawn mowing? Possibility, but again, its not the same as a regular job, even to match part time hours, you'd have to find 8 or more people a week willing to have their grass cut, and maybe return in the summer once every 3 weeks? That's 24 different households in a community.
They are 16, this is the longest summer of no responsibilities and no work that they are ever going to have.
Let them enjoy it.
Oh and the killer time for applications to supermarkets and other traditional employers of young people is late August/early September, just as the kids that are off to uni start to leave.