Utility of money is just how much value one (an individual or entity) can derive from having x amount of money. Utility and value might be defined by self referencing each other but that's true for many other concepts or words in a language for that matter. It doesn't make it problematic or unscientific.
The point is that it matters how much certain amount of money or wealth is worth for you not how much wealth you have. It's an important point as it explains why calculations purely in money terms are useless for making financial decisions (as seen in example of insurance)
It's not like that at all.
If you want to argue that one can't define value then I am out.
If you admit "value" has well understood meaning then utility function of wealth is just how much value one can derive from it. We usually measure wealth in money for convenience. I really don't know why you are so against it. There might be a lot of nonsense associate with it but the concept itself is pretty simple and useful.
The point is that it matters how much certain amount of money or wealth is worth for you not how much wealth you have. It's an important point as it explains why calculations purely in money terms are useless for making financial decisions (as seen in example of insurance)