Learn English – In the native language, we have a saying – a stone will get a wretched person, going uphill

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.

I believe the idiom have an eye to/for the main chance comes close. It is more of a British idiom.

Someone who has an eye to/for the main chance is always ready to use a situation to their own advantage. Cambridge

As you can see from the definition, the expression emphasizes that one is always ready to take advantage of a situation and always looking for opportunities. He is basically opportunistic.

The origin of the phrase comes from gambling:

The origins of this expression lie in the gambling game of hazard, in which the person about to throw the dice calls out a number between five and nine. This number is called the main or the main chance, and if someone rolls it they have won.

_{Oxford Dictionary of Word Origins by Julia Cresswell}