What happened to parents paying their childrens' way, then? Or, in
teenagers' cases, a paper round or Saturday job?

Not everyone in London is middle class. By providing the travel
concession
as a universal benefit, children from disadvantaged backgrounds get to
travel free without the stigma of receiving a targeted or means tested
benefit. There are other positive externalities to getting young people
used
to using public transport on a regular basis which include the
environmental
impact.

Is the concept of getting a job to pay your own way a uniquely "middle
class" one, then?

No, it is uniquely middle class to begrudge people who are less fortunate
any assistance in bettering their situation. Lack of access to public
transport is one of the main causes of social exclusion. By providing free
transport, access is given to education and other services (without stigma)
and it ensures that money earned "to pay your own way" can be used to for
things like food and heating.