I'm admittedly struggling to understand how he's drawing this conclusion, as I think he's equating utility with expenditures? If so, this is completely wrong. Your first few thousands of dollars are used for things like food, shelter, and basic healthcare, and they generate massive utility. It isn't constant.
But I also take issue with the first assumption, that each hour of leisure time provides constant utility. Is that true? Does a poor person get more marginal value from that last hour of "leisure time" on the streets? What about poor people who go to bed hungry, are they doing that because they're rationally basking in all these extra hours of leisure time?
There are many reasons people are poor, but people choosing to be poor as an economically rational decision sounds crazy, and that's because it is. Just measure the happiness of the non-working poor vs. people making 30k, and I think you can put this one to rest.