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.