I empathize with the OP. I wanted to post a reply to probablyfiction explaining to him. Instead I will list the reasons why burnout could have happened to him.
Point 1 "unless X has a toxic work culture,it's likely the author brought it on themselves by pushing too hard."
An overstatement unless you know what he faced. You were hired for your problem-solving ability where pushing hard can be expected at times. In reality this could be he's not pushing too hard so much as the existing environment which leeches effort and adds burden to every step. Toxic culture is somewhat unlikely but what if he inherited a huge technical debt, this happens a lot.
Point 2 it's a terrible feeling to have to spend your most productive hours working on someone else's problems. He or she may find themselves drawn toward entrepreneurship
Wrong. IMO you can either work as a product engineer role building something new (hopefully) or as support role integrating products/fixing glaring bugs crappy code/migrating systems. The latter is not exciting. A developer MUST be capable of both and SHOULD be given at least an opportunity for both. It does not have to be in your own startup. I maybe a coder but I am not a horseshit shoveler.
Point 3 Office culture is draining. Some people aren't cut out for it.
Why ? It doesn't have to be. Even then why assume work culture is the culprit here and he is not cut for it.
Point 4 Additionally, programmers tend to push themselves too hard for too long out of a misplaced sense of loyalty to the company. They don't see that the work they do is extremely valuable in and of itself.
Translation the coder has to see the fine line of enthusiasm between "working already" and overzealous. Seems like a hypothetical problem unless you tell why is it relevant here. About the "work" you say, you should tell what "work" is. It is not just the stationary lines of code lying on repository, but the amount of effort that went to write it, debug it and will go to maintain it. A badly written project demands higher effort to keep it running than what you could imagine.
Point 5 a coder doesn't owe the company anything more than their agreed-upon hours of work.
I know that is the agreement. But I don’t know if you have seen places where boss wants work in a deadline and not hours a coder spends and wants it done. You should remind him that then.