You are correct. I don't think Americans want an escalation. I think Americans are trying to follow the old model: Let Israelis do killing nth time the number of Palestinian killings and be done. But Israeli situation is precarious: If they let Hamas stand, however weaker, then Israelis have lost and Netanyahu is toast even sooner than he would be otherwise.
Why wouldn't Russia? Supplying through Wagner provides plausible deniability of Russian role.
But Israelis are in a 'damned if you do, damned if you don't' situation. They can't let Hamas stand in Gaza and they can't ignore American commands for long. Remember: America is a Superpower with its own interests and who knows what MBS told as some 'redline' to Americans? The GCC Arabs maybe fearful of Tel Aviv/Washington but they can't ignore their own 'street' if this become a regional war. They may face President Najibullah treatment by the Taliban!
Because, after years of supporting the Palestinians, the GCC Arabs are okay to have 'peace' and prosperity in their countries instead of asking for a destruction of their countries by Israel/its backers. And a lot of the GCC rulers 'own' is in the Western financial institutions/countries: Their yachts, their mansions, their stocks, their bonds, their real estate, their $$ reserves, are only a few days away from Biden's flick of his pen. Happened to Russia recently!!