> I guess the president is democratically elected too.

That's a bug, not a feature though.

What happens in practice is that you get president and government from different parties and they carry their political fights on instead of governing. The president vetoes the government just for the sake of sabotaging it for his parties political gains. We've had already multiple instances of president and government having almost independent foreign policies, crazy. Trump, e.g., has de facto been ignoring our prime minister Tusk (which by constitution should decide our foreign policy) and has been talking only with our president. And that's not on Trump, but our crappy Polish constitution for allowing this mess.

On top of that, governments are based on holding a majority votes, presidents just stay there whatever happens and whatever chaos they create.

I think that there should be a politically, democratically elected government whose job is to make laws, and the rest of institutions should be as bureaucratic and boring as possible, with as little political involvement as possible.

Again, I really like how in Italy or Germany the president is not involved into law and policy making. Ever noticed how all the countries that deviate towards autocracy...are all presidential based?