You could just say that the money has no value by itself, and the money an individual has represents the amount of goods that he can have. Thereby you can exclude money out of the equation completely and would lose almost nothing from the picture. If we consider value as utility, then money has some utility to us by and of itself, since it gives us the possibility to choose what to buy (and what not) as well as postpone buying altogether. Whether this utility abides by the law of diminishing marginal utility (i.e. diminishes with the amount of money an individual has) remains to be seen...
Personally, I don't think that it does, therefore there is room for doubt about the nature of this utility
I think it does, but since money can purchase almost everything, and human's demand are so many, you are actually asking wether diminishing marginal utility applies to all the human's demand added together. That is much more complicated than asking the same question for a single utility like bread or milk
In fact, even with 100 utilities, you might run into a problem that the demand appears to be unlimited due to you have less time to consume: Even if you spend all day playing games and watching new movies, you might end up with missing out more and more games and films, not because you don't like them, just because you don't have time (And generally economy theories are not working any more in an environment that has abundant supply)
And you can not exclude the money out of the equation. Without money, you will lose the means to access any other utilities. This transaction demand is also what gives fiat money value, and it enabled most of the commercial activities
Without merchant acceptance, the money is like you described, has no value, but the universal acceptance itself, although does not generate any utility, injected value for currency temporarily until the next trade happens
A good example: In 2009 North Korean suddenly announced that the old currency will not be allowed in circulation, thus anyone holding the old currency immediately lost most of the purchasing power of those currency. The value of those currency suddenly disappeared because the acceptance is removed nation wide
One bitcoin private key without blockchain is totally useless, but once it is in the network, can be transferred between people, some people will start to accept it as a payment method, then it start to have its monetary value. Similarly, fiat money without merchant acceptance is also useless