Why a chef is the boss of the kitchen but a fullstack dev is not often the leader of a dev team?
IMHO the comment is a bit misleading, in the sense that a chef is the boss of the kitchen mixes up and/or attempts to compare the unique leader/chef role (in French, chef literally means chief/boss/leader) with the expertise required to fill that role. By contrast, the full-stack developer is just an expertise, it is not a unique role.

If I'm not mistaken to become a chef one has to have experience in every other position in the kitchen. Similarly, a full-stack developer's expertise covers frontend, backend, etc - every other more specialized expertise required for the team's product. Which is probably the starting point for the comparison in the comment. 

As I mentioned in my answer to that post, scale matters. 

In a large team, the requirement for the leadership position isn't necessarily the technical expertise. Which is why often in software development the team leadership's role is a management position, not a technical one. Clear labour division example, if you want.

It's not impossible to even have multiple persons qualified to fill the chef role in the same kitchen, but only one can actually fill that position at any time, a chef in this context really refers to the role, not to the expertise. Also - there may be many types of chefs in a kitchen, see, for example Chef Jobs on Cruise Liners :)

But in a small resource-starved team where the team leader has to be selected from between the only full-stack developer and one or more less-experienced ones - probably the full-stack developer will be the leader, just like the only person qualified to be a chef in a small kitchen.