Two-party system is the enemy of democracy to begin with

But maybe even they would be more inclined to do better everyday

Many countries actually have such systems in place today, even Russia (lol) - not that they work too well.

Normally, there are two sources of issues here: petitions can in fact be declined, and, in cases where the signature count depends on scale of the petition they can be intentionally escalated as to make it impossible to gain enough signatures. Besides, in many cases petitions can be left unanswered for longer than promised.

Long story short, the system is open to shenanigans and doesn't make the government truly accountable.

We need the system that would actually make politicians rapidly lose their jobs when they ignore public opinion.