You're changing your argument because your account of history is wrong.
Hamas got into politics, same as the Ira of Northern Ireland did with Sinn Fein (which after a few years turned its back on warfare to dedicate fully on politics)You were firstly suggesting that the Provisional IRA entered politics via Sinn Fein when the truth is that Sinn Fein, one of the founding political parties of the Irish Republic, entered into terrorism through the PIRA.The movement of Hamas into politics was a massive step towards the right path. As I said before that is exactly what happened in Northern Ireland and after a few years of struggle the resul is this. peace in the north. Sinn Fein is now a normal political party that refuses violence as an answer, shook hands with the queen and all that jazz.
Secondly, you suggested that the political processes caused the PIRA and Sinn Fein to ultimately end their campaign of violence. This only true in as far as the political process was going to exclude Sinn Fein entirely. The PIRA was still actively engaged in terrorism while early negotiations for the Good Friday Agreement were underway. The British and Irish governments and various Republican Irish parties in Northern Ireland and the Republic decided to move on without Sinn Fein. So, faced with the choice between continued struggle and total political irrelevance or disarmament and political survival, Sinn Fein chose political survival.
So, this is a completely different narrative. A path to peace was found in Northern Ireland by refusing to engage with people who used violence as a political tool and looking for other partners that were ready to move forward. If anything, the lesson from Northern Ireland for Israel is to keep killing Hamas and refusing to engage with them politically while looking for new partners to start talks with.