The company I work for is Dutch (very egalitarian, consensus-based work culture), hires only people with academic degrees (love to argue), has a very flat organisation (very little hierarchical power to make anyone do anything) and on top of everything else, it's a consultancy with a software development branch at heart, and the programmers are told the same thing as the consultants -- be visible, have an opinion, make it known, and so on.

So this is completely typical for every single ticket and the team lead is powerless.

What works is: let the team know the planning, make the whole team feel responsible for making the planning, ask this programmer to make this ticket happen, let him come to the same conclusion as you regarding the planning (argue), and then have him figure out how to solve it in half a day. He'll probably also add the private variable, or come up with a similar way.

Alternatively, add the damn private variable yourself, if it's only a single line.

Trying to force people to do things on a level of detail that is well within the area they want to decide themselves doesn't work.

"Team lead" isn't "team boss".