Only the US was so rich and strong it could afford to indulge in the impractical.
I'm not sure how impractical it was. By the time the US backed Israel, Egypt, Syria, and Iraq were already firmly in the Soviet camp. Israel made sense as a counterweight, even putting domestic politics aside.
Yes, there was Jordan, and Saudi Arabia, but Saudi Arabia in the 60s was still quite weak, and Jordan was still more British-aligned than the Americans cared for.
A "fourth option" might be a separate non-aligned organization. The non-aligned movement was very anti-imperialistic. I suspect that many countries (most of Central and South America, for one, maybe Thailand, Pakistan, maybe some of the neutral European states) might be tempted into a bloc that's neutral but not anti-West.
Israel was founded by Europeans. I don't remember the source but English and German were more commonly spoken in Israel then Hewbrew in its early years. I even think French, Russian, and Polish were common too.
It be like if Australia or South Africa pre apartheid joined the non-aligned movement. Technically they could, but if the Cold War was a European people's based conflict why would European settler nations want to miss out on all those juicy weapon deals?
Israel was founded by Europeans, but within a decade of foundation, Middle Easterners were a majority of the population. Plus, Jews weren't considered to be particularly European by the Europeans, and their history as an oppressed people throughout Europe could have been made to resonate with anti-colonialist factions. Certainly the Israelis didn't consider themselves to be a European settler nation.
Even outside of Israel, the narrative of Israel as a European colony wasn't very popular/common in the period being discussed.