Israel isn't "annexing southern Lebanon". Israel already controlled southern Lebanon and withdrew. Even recently Israel was deeper in southern Lebanon and withdrew - and is now paying the price for that. Israel was already in Beirut .. and not so long ago ( https://en.wikipedia.org/wiki/Siege_of_Beirut )

Israel is pushing back Hezbollah that's attacking Israel's north. Hezbollah decided to join the war and it's firing at Israeli civilians and towns with statistical weapons (rockets).

It does seem like it's at least some sort of short term win for Israel but it remains to be seen what the long term game looks like.

> how important the popular support even is

To see the effect of losing popularity, see how AIPAC's power in the Democratic party has begun to wane following their defeat in New Jersey.

A common mistake those deploying money in politics make is forgetting that the endgame is votes. The money helps buy votes. But if you're losing votes, you're losing votes.

> right now, the state of the war is a win for Israel

If hostilities end right now, yes. There is zero indication that endpoint is proximate.