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.
This is irrelevant. As long as utility is monotonic (more consumption => more utility), the conclusion holds.
But I also take issue with the first assumption, that each hour of leisure time provides constant utility.
The blog post only assumes utility is monotonic in leisure (i.e., more leisure => more utility).
...measure the happiness...
How do you reliably do that? The only thing I'm aware of which comes remotely close to this is revealed preferences.