There is, hypothetically, a market price for these jobs. It is below the minimum wage (otherwise there isn't much point in the minimum wage). It is set by the pairwise negotiations of companies, customers and employees.

If there is an avenue for the market to reach equilibrium (eg, by hiring children cheaply) then the market is going to try and take it. It will just create a class of jobs that adults can't get employed in because they cost too much to hire.

It isn't a question of what the kids would like, it is a question of what choices consumers and employers will offer them. If wages were set solely by employee's incentives then they'd all be paid much more than current market wages.

Do the kids become more incentivized to work if their parents income decreases?

I would say yes

Result: a class of families consisting of unemployable parents supported by the state and their own working children :(

I would say maybe it’s not a dream society if we’re creating a class of helots to maximize teenage labor force participation.