I would argue that political violence was at most times “widespread” in Roman politics - and other ancient societies as well. Now, Rome can be seen as the first or one of the first societies which had phases of less political violence and orderly transitions of power (during republican era and other times).

But even early founding myths of Roman societies have violence as an option. My following hypothesis is based on the works of Marie There’s Fögen (Das Lied vom Gesetz, 2008 and Römische Rechtsgeschichten, 2002).

The stories of Lucretia and Verginia both feature situations in which the people rose against tyranny and - cleary by means of violence - re-established the rule of law and decency.

Therefore the option of reverting to violence in times of crises (in extremis) seems deeply embedded in the cultural imprint of Roman society. Keeping in mind the strong martial tradition and the connections between republican voting and service in the armed forces this is maybe not surprising but can be easily forgotten it one only looks at the calmer times of Roman history.

Now regarding the Gracchi brothers, the crisis which had befallen the Republic after the second Punic War may certainly qualify as an in extremis kind of situation. But the Gracchi were hardly the ones introducing violence. Let’s not forget that Tiberius Sempeonius was murdered by the senatorial elite. He may have broken constitutional law but the ones resorting to violence were those reacting to the first populists.

If you question how violence became more common - I would coin it more common again - after the 2nd Punic War I would argue that the main reasons were the deepening economic divides and social developments in Roman society. On a broader scale you could also argue that a society largely benefiting from violent conquest and plundering which is also establishing a never before seen kind of administration in this region of the world (Empire) is actually maybe quite unlikely to stay as law-abiding and peaceful as a regionally limited city state with mostly agrarian citizens.