That's like wiping out the Democrats after an attack by General Kerry of the DNC, but leaving Republican infastructure alone.
Hizbollah is an integral part of the Lebanese government, they hold 14 of 128 assembly seats, sypmathizers better than double that. They serve in the cabinet. The government not only allows Hizbollah to maintain a private army in violation of international agreements (and Security Council resolutions), but they allow supply through Lebanese infrastructure. Ports, airports, roads. Hizbollah is given free reign to maintain order in southern Lebanon (and South Beirut).
This wasn't an attack by a terrorist group based in Lebanon, it was an attack by Lebanon. You can no more separate the two than you can separate the Taliban and al Qaida
Yes, you can attack Hizbollah by attacking their strong centers in the south and Beka Valley. You can also send in SF to find the leaders and take them out where ever they are. The smart thing for Israel to do is to say "We do not declare war on the Lebonese people, we declare war on Hizbollah and those who protect them."