An English idiom for "solve a problem that has been solved"?

The standard English expression for solving a problem that has already been solved is...

Reinventing the wheel
duplicate a basic method that has already previously been created or optimized by others

The idiomatic push at an open door has a completely different meaning...

to achieve what you want easily because a lot of people agree with you or help you (usually in continuous tenses)
The campaigners are pushing at an open door because most local residents support their campaign against the new road.

A phrase that would apply to a person who is arguing a point that everyone already agrees with:

preaching to the choir:

To commend an opinion to those who already accept it.

The phrase comes from an earlier one, preaching to the converted (1867).

The idea has also been expressed in another phrase that refers to an unnecessary act, that is, kicking at an open door.