Price/money is useful but the goal is to create value for consumers, not to suck them dry :) Profit is a market signal/reward for producers but that doesn't mean that it's == to value created for consumer (see my reply above about monopolies, externalities, etc.)
"A startup is a small company that takes on a hard technical problem."
The monetary value of "discretionary time" is the amount of money you could have made if you did something else instead.
More specifically, you can't always do something else revenue-generating instead, and nor would you want to. What's the value of spending time with your children? Can you see how some people would not accept any amount of money to displace all of that time, whereas some others may would (in cases of paid-for adoption / slavery)?
I think your outlook is lacking in a specific economic concept: utility.
Obviously, you can't be doing something revenue generating all the time. But for determining the value, that's the way it's done.
When you choose to hang out with your kids, you choose that over everything else you could have done. One of the things you could have done is earn money. You decided to give that up, so the monetary cost of doing what you did instead is the money you could have earned but didn't.
The market value of your time is, assuming you have a job or charge a consulting fee or whatever, is already determined.
If there is no price level at which a transaction will take place because the supply side values the utility of the commodity higher than any amount of money the demand side offers, then you cannot price it in terms of money.
What's the value of the Mona Lisa? You can't just go off and find some other painting(s), look at those prices, and come up with a number. The only way you can discover its market price is by testing the market: i.e. putting it up for sale and inviting bids.
A person's time is not substitutable; one hour in the morning is not equivalent to one hour in the middle of the night; and 24 hours of it every day is not equivalent to 24 times the price of 1 hour of it. It's not fungible. This limits your ability to put a monetary price on it.
"Why is money the sole metric of wealth."
is a) ungrammatical and b) basically a tautology. money is the sole metric of wealth because wealth, quantified, is money.
not trying to detract from the point you raise, just getting a bit OCD about logic and grammar.