That's not the same as controlling for socio-economic status. A millennial living in a city-center apartment and a boomer living in a paid-up house in the suburbs might have the same income, but they have very different life circumstances and experience very different levels of stress.
They also controlled for "age, sex, annual household income, economic status (eg, employed, retired, or unemployed), highest educational qualification, and couple status".
Taken together those are likely to capture socio-economic status in all but a few edge cases. With a very large study size, and a huge result (30% difference) it's unlikely that it can be explained by socioeconomic status.
If this were a small study or found a small result, then edge cases like lower income millennials with no degree who live in inherited houses in wealthy areas with trees might skew the results.
If you disagree with the studies methodology fine, but I'm satisfied that authors have done their due diligence with respect to socio-economic status.
The problem with that is tree coverage drastically goes up again when you hit "broken washing machine on lawn" levels of poverty due to just the general amount of old growth in rural poor areas