Nobody has ever found a better way. What you're willing to pay for something determines its value to you.

The assumption that price paid == utility is only valid in the economic equivalent of "assume no friction" from intro-level physics. The equivalence isn't guaranteed to hold once you bring individuals' budget sets and various market distortions into play (information problems, sticky prices, et alia). You see the same thing on the supply side, with the assumption that price sold == marginal cost breaking down in general, except under very specific conditions.

Bottom line: you're still only valuing something by what you are willing to pay for it. That is its value to you. (Others assign a different value, but that's how the market works.)

Supply + Demand determine the value.

Right but notice utility is mentioned nowhere. Demand is totally irrational, because it's human. See toilet paper, fujifilm x100v, tesla stock, NFTs. Supply can be arbitrarily subject to the irrational whims of humanity as well - see oil, diamonds.

"Supply + demand" seems an oversimplification of a huge chain of irrational human links.