At end of the day it's objective reality that none of Saudi Arabia's US supplied modern weapons will work in a conflict with Israel, e.g. their F-15s would be unable to target Israeli jets assuming they can even take off.it is still single most important country but it does not mean it can take alliance of much bigger population countries alone
and yes Saudis are independent you can see from there stance on China related issues. No one can sway them. so your earlier assertions of too deep (meaning of too deep) in Western Camp were wrong along with defeating Houthis.
Houthi lost the country despite Saudis not properly closing the supply lines and now getting bombed by some one else. Thats the Saudi power to hand over problem.
Even UAE is worth more than Iran.
Saudi Arabia is independent in so far as playing their oil card, but when it comes to war they're completely subject to what's permitted by the US and Israel.