Taking out Hizbollah would ipso facto benefit Israel. We’d need nothing from Putin except his covert OK to operate in his turf, Syria. And he wants them gone, too. I don’t see your reasoning.
Hezbollah's not the largest threat to Israel, and Israel currently has leverage that will be gone (for a short while) if it destroys the Hezbollah bunkers. There's no rush to get that done, unless Hezbollah acts first. If that happens, the bunkers go, then Russia will have to get rid of the Iranian infestation or else.