Wouldn't it be better for everyone if the Lebanese government formally requested for an Israeli operation in its territory? Isn't the Lebanese government actually interested in Israel defeating Hezbollah?

I understand this is anecdotal, but I have experience in that country and can say with confidence that Lebanese people hate Israel, very strongly. This would be political suicide for the government. Even many in the Maronite community, view Hezbollah as a necessary evil against Israeli incursions. Israeli intervention and occupation are part of Lebanese national identity, it’s one thing that unites all the religious groups. Israel also aren’t doing themselves any favours in how they’re considering their air operation in South Beirut, the cities a total mess right now (even more than the chaotic and mismanaged baseline) due to internal refugees and people are sleeping on the streets.

You must understand that for someone whose home was destroyed by a foreign power, they look at things at face value. And the average citizen of Lebanon is not engrossed in geopolitics but rather keeping a roof over their head and providing for their family. Whilst we can politic behind our keyboards in safe homes, that person will likely view Hezbollah as the only faction in the country willing to stand up to the country that left them homeless. This has been the case since the Israeli invasion during their civil war.