If we exclude the odd snickers about cultural taboos (ahem ahem), you cannot in good conscience deny the privilege Asians have compared to some other minorities, pandemic assaults notwithstanding. Many white supremacists/nationalists admire Asians and acknowledge their work ethics and intelligence. I can't remember the documentary but for what it's worth, a white-nationalist was making the point that he doesn't care so much about supremacy, but white nationalism. He was in many ways Asians are even superior to whites (again take that FWIW because i don't remember his name to quote here).
I was picked on in my all-white school for my curly hair and darker skin but I never really experienced true racism until I dated a black guy. He played for the Toronto Argonaults making a lot of money and dressed nice, yet cabs would not stop for us once they looked past me and saw him. Waitresses would him to pay up front for our drinks and being surprised by the big tip. Sure, things like 'where are you from' bothered me when I grew up in Canada but I was shocked by the life he had to endure even though he was an educated, high-earning black man.
I wouldn't say there is a hierarchy, but there are different branches, and indeed Asians are on a branch that's not as affected as some of the other branches.
To say it's privilege is not incorrect IMO. They are more likely to get that phone call for the job interview because it's assumed they're smart. They're likely to get that apartment rented to them because it's assumed they're quiet and will pay their rent on time. They're more likely to not to have their hair products locked away because it's assumed they might steal. They may get slighted for being just the waving neighbour who doesn't socialize, but hey.. at least they're quiet other than the sound of their kids playing the piano.
I certainly see the differences and how they can be seen as privilege, and I don't really think a person of European ancestry who has not experienced racial injustices has any room to discount their experiences and/or mansplain why they're wrong.