If you have given up on trying to grow food this year.... Why not simply use a strimmer, or just run over the garden with the mower?
While there is some slight elevation difference, really doubt that a mower would have a problem.
With 5 children, I wouldn't expect to have a perfect lawn anyway.
Save the effort for something that matters.
Food is pretty important, why not harness all that kid power, and turn the soil with a shovel, and plant something edible?
Might be a great learning experience for the entire family!
In all honesty, the existing vegetation doesn't look like any big deal.