It's not quite another correlate of wealth. What they found was that resisting the marshmallow was correlated not to being distracted from thinking about it, but to doing something that helps with the frustration.
Later data review showed a correlation with certain life outcomes, and further study showed that nurture played a bigger factor than biology and finally that the marshmallow isn't a predictor of life outcomes.

Money and the things you mentioned are one of the factors for better ability to lessen feelings of frustration. But so is coming from an Asian household, but only if the marshmallow isn't in a box. People from European cultural upbringing do better when the treat is in a gift box.

Getting dismissive of the study makes you miss out on what they were primarily trying to learn about: how do you successfully resist temptation?
All the rest was just a side effect of normal number crunching after the fact. They knew they had a biased sample of kids from the beginning since they were all sourced from the Stanford on prem daycare. They didn't expect that to have much impact on what they were actually studying. And then it took ~20 years for follow up studies to finish, since the kids need to grow up after eating their marshmallows.

(Asian heritage households reported that they tended to have food visible to children that they had to wait to eat more often than European heritage. European tended to have gifts be a special occasion where the norm was to wait to open them, where Asian had more instances of common, small gifts given and immediately opened. So one set simply had more practice not feeling frustrated about seeing food they couldn't eat, and the other with waiting for permission to unwrap a gift)