Considering how USD enters the market, is the USD actually unbacked? Could the USD represent a unit of a “fund” of securities held by the Fed? From my understanding, all new money enters the market either 1) by the Fed lending it to banks, so the Fed essentially owns bonds in the exchange, or, 2) The Fed purchasing other financial assets. So for each new dollar, it is represented by some new security in the Fed's "vault". This doesn't sound like it's unbacked to me. Hypothetically, if the Fed and the dollar "went under" would there not be some obligation to distribute these securities among all dollar holders?

Furthermore, what is stopping the Fed from acting like a fund where each dollar represents a (very) small percentage of all securities in "the vault". To me, it seems like this would create a basis for value of the USD. If a USD-holder were able to trade in his dollar for tangible securities, would this not increase his confidence in the system itself? What's the reason the Fed doesn't operate this way?

Very interested in hearing more educated responses to this, thanks in advance.
When the Fed purchases something, it creates the money to do so out of thin air (it just changes the balance in a bank's Fed account. Click, click.). Similarly when the fed is paid, the dollars disappear. From the Fed's perspective, dollars are very ephemeral and not backed by anything.

Dollars are not like debt. They don't entitle you to anything the Fed owns. The Fed can't "go under" in the normal sense of word (bankruptcy) because it has no obligations to repay or if it does, it creates the money to pay those obligations with no limitation.  It goes under only by an act of law.

It would be possible for the government to abolish the fed and take its assets, but that doesn't sound much like a fund to me--there is no reason to think those assets, whichever ones remain meaningful in that context, would be disbursed to dollar holders.  I would expect them instead to be kept by the treasury and used to reduce the national debt.