Why in the world would you expect them to? Evicting Hezbollah will get Iranian violence targeted at them rather than simply being incidental. (Same as neither Lebanon nor Syria shoots at Israeli aircraft. They know Israel could smash their air defenses if need be, but that Israel isn't going to start something with them.)
We even saw it with Europe--European governments would tell terrorists that it was getting too hot, move on before we have to come arrest you. That's why the Mossad was assassinating the Munich terrorists--they would always be tipped off and run before they were arrested.
I think there’s been a bit of a misunderstanding about what I’m suggesting. My argument isn’t that the Lebanese government would take bold direct action. Instead, I'm proposing that Israel continues to weaken Hezbollah through targeted military operations, creating opportunities for the Lebanese government to regain control incrementally.
The idea is that Lebanon wouldn’t necessarily confront Hezbollah head-on but rather "step aside" while Israel handles the heavy lifting, as you put it. Once Israel severely diminishes Hezbollah's influence or infrastructure, the Lebanese government could move in and reclaim areas with little resistance—with Israel turning a blind eye. This way, Lebanon could gradually stabilize without overtly challenging Hezbollah themselves. It’s a strategy based on plausible deniability, where Lebanon wouldn’t actively engage Hezbollah but could still benefit from Israel’s efforts.
In short, it’s a cooperative dynamic without explicit collaboration, where Israel’s military actions would indirectly enable Lebanon to restore order when the moment is right.
