How to set "Advertising Features" and other settings through Google Analytics Management API?

How can I set this and other values through the Management API, or other means?

You cant the management api is very limited in what fields you are able to update. Unfortunately unlike a number of other google apis documentation it doesn't actually tell you which fields are writeable. I will mention that to the team.

If only a small fraction of the settings for a property can be set through the API, that seems kinda useless for managing large accounts, as the docs claim you can do - so I assume there has to be a way I'm just not seeing here.

The API was not intended for managing large accounts IMO. It was probably only intended for getting information about what accounts a current user has access to.

What i suggest that you do is create a feature request asking for the ability to update the fields that you are trying to update. The team is normally very open to those types of requests.

Link it here and i will be happy to bring it to the teams attention next time i speak with them.

AFAIK this and many other features are not available in Management API.