I see a lot of description about what must be done by a Code Steward, but I'm having a hard time seeing what benefit the CS themselves gets from the process.
Topic on Talk:Code stewardship
I think that's a fair question. Although I don't necessarily see them as sticks, it wouldn't hurt to list out some benefits. That being said, the original context for defining CSes was to ensure that those components that WMF was dependent on in production were properly resourced. Hence the "Responsible Wikimedia Team" column on the developers/maintainers. Once I dove into it a bit more, it seemed like "stewardship" was a better fit than "ownership/responsible", and that there was no reason to limit the stewardship model to WMF teams or to code that was used in WMF production.
One of the interesting discussions that I had with Kunal was the need to have a two-sided contract when it came to SLAs. Requiring certain activities to happen prior to commits (i.e, linting, testing, etc...) before the stewards would perform code reviews. So in a way, being a steward provides a mechanism to encourage behavior by offering up support provided the contributor meets a set of steward-defined prerequisites. These prerequisites can be existing global coding standards or steward specific.
That being said, there'll likely be some prerequisites to being a steward, like current voting status (+2), etc...
I also don't think all code has to have stewards. If the code in question is important enough to a person or group and they depend on it in anyway, it'll be in their best interested to be a steward and establish more rigor around the development and support of the code.
I'm not sure if that additional context helps at all and/or give you any ideas about explicit benefits, but I'll think more about some benefits as well.
Only vaguely related: User:AKlapper (WMF)/Code Review lists benefits of code review as "knowledge transfer, increased team awareness, and finding alternative solutions. Good debates help to get to a higher standard of coding and drives quality." (and also lists all the related problems to face).
But that's of course just the benefit of having such a process, not the benefit for the stewards themselves.