You're 1/3 correct.
Israel did receive only limited backing during the 1948 war of independence/conquest (recognition probably the most important + loans also), but it received quite strong support during the 1967 war (support was close to euphoric after), and massive support in the latter stages of the Yom Kippur war and directly after.
In fact, there is an open question as to whether Israel would have recaptured the territories had it not threatened a nuclear response in order to bring about a huge shipment of weapons from the US (and other countries), which it received.
Also, the Yom Kippur war would probably have been avoided had the US
not embarked on a course of Israel-first ("stalemate" - i.e. support for Israeli expansion indefinitely) during 1967-1973 - based largely on racist contempt of the Arab states, US/Israeli economic interests, power politics etc.
Though it is wrong to say that:
"The U.S. has financially and militarily supported Israel since the beginning to such an extent that Israel wouldn't exist otherwise",
it is not wrong to say that U.S. financial and military support of Israel is unprecedented in scale and character, and Israel would probably (maybe very likely) have been destroyed at some point had such support not been in place.
Certainly that is the case during the 1967 period until today, as recognised by pretty much everyone. For two minor examples, Israel would not have been able to expand into the occupied territories without American support, and nor would it have been able to direct its forces towards the north had America not bought Egypt off with client-state status (in the form of military assistance, aid, diplomatic contacts etc).
Though you could argue that such actions
reduce Israel's security in the sense of pre-1967 Israel, that is not the Israel of post-1967 or today. That (expansionist) Israel would have been wiped out had it tried to pursue those policies without American backing, just as it would no longer be the "most powerful in the Middle East", and "stronger than ever", without such backing in the present (in fact highly vulnerable, given technological developments).
All this is of course putting aside actions which the US takes of its own accord which help Israel as a predictable side-effect; i.e. direct consequences of the "special relationship" - say, the US invasion of Iraq (eliminating the "Eastern Front", eliciting euphoria in Israel), sanctions on Syria, pressure on Iran (also acquiescence to direct Israeli inteference in Iran, also supply of "special weaponry" etc). All quite substantial.
There are also things such as tacit agreement to allow Israel to penetrate (more accurately: rob) the Iraqi reconstruction program (co-ordinated at the highest level, suppressed for fear of a "political explosion" if the facts were known) etc. All possible thanks to the US system of domination in the region - support for Egypt and Jordan being the most important (the latter the "sheep-dipper" of Israeli products to be sold in Iraq, to use military terminology) - and vital for the Israeli economy, particularly during the current Intifada.
As for Israel "going on to do its own thing", you're half right. It is doing that right now - causing the US considerable discomfort with its covert operations in the Kurdish areas of Iraq, inflaming Turkey, Iran, most of Iraq and others. That's because Israel views U.S. strategic policy towards post-war Iraq as contrary to its interests and it is therefore committed to destablisation - which is not the U.S. goal, but there is little Washington can do about it for the moment.
The reason for "half right" is that Israel does not have the room for independent action you seem to be ascribing to it. To note a minor example, the U.S. is currently damaging the Israeli pharmaceutical industry quite significantly in a dispute over patents (the largest producer may leave Israel in fact). The Israeli lobby is nowhere in sight, and there is nothing Israel can do about it, because the patent move is favoured by powerful interests in the United States, which don't care about Israel one way or the other. In this case, Israel cannot go on to "do its own thing", because its entire economy is artificially based on massive foreign investment and capital, shored up by US aid, credit loans, and bank guarantees.
Though that is a domestic example, the underlying principles extend to foreign affairs.