It's unsurprising that people for whom the system is likely to work a bit better than average are, on average, a bit more sceptical about changing the system. But, it's good to have numbers to back up the intuition.
It’s not the system, it’s people as such. Being attractive confers benefits pretty much universally. That should come as no surprise for a species that has been through millions of years of sexual selection.
Otherwise, you are correct, the fallacy is to assume what works for you will work for others.
Another story could be that if the system is working for you, you're probably getting dental, can afford skincare, can splurge on the nice barber, etc.
Attractive people can roll around in mud and refuse to shower for a year and they'd still have plenty of suitors after them. But cosmetic factors can elevate otherwise average looking persons.
They did "control for socioeconomic status", I assume that means levels of costly self-care were distributed across both political aisles when they were setting up the study.
