Agree with other posters that privilege does not feel like the right word for this.
I've been a size 8, a size 24 and am currently between a 14-16. My weight has influenced the way I've been treated, but regardless of size it's not always been positive. But the judgement on my perceived character and behaviour has been consistent - the only thing that changed when I was at my heaviest was the assessment. I was fat, so I must be lazy, unmotivated and selfish ("people like you are why the NHS is struggling").
One of the ironic benefits to being heavier, is that the volume of sexual comments noticeably tailed off. They were replaced with comments about how I was too fat to fancy etc. But I wouldn't say that the comments when I was slimmer were any better. Personally I didn't like strangers telling me I'd got great tits, or that my bum needed someone's cock in it.
Mixed bag at work as well. I wasn't deemed suitable to be photographed for the PR material any more, but I was considered to be a solid safe pair of hands for getting high quality technical work done (previously "You need more experience Elderberry"). Which is strange because the turnaround in the assessment of my competency took about 18 months, and the only significant change in that time, was a 3 stone weight gain.
The combination of being middle aged and heavier has now made me largely invisible, presumably because I've reached that magic combination of being too old to be of any sexual interest or worthy of many comment, even an insulting one. Which is lovely to be honest, because I can get on with my life and not have to face a constant barrage of comments on my appearance. The few comments I get these days tend to be from young women and are fake-sympathetic ("You'd look so much younger if you got highlights and covered the grey"). I don't mind them because what they don't know is that I like my grey hair and I don't care about anyone else's opinion or judgement of how I look. I know they don't know this, because I thought exactly the same way when I was their age.
It does however, make me feel utterly furious about all the shit girls and women have to put up with to get to this point. So my take on this would be that instead of "skinny privilege" why not rail against the actual cause of the problem, which is men who can only view women through the lens of their fuckability?
It's this that sets the societal expectation that we judge ourselves and others against: Our slimness, our "prettiness", have you shaved, are you tanned, is your hair colour up to date, are your nails done, what about make-up but not too much, are you "fighting the signs of ageing" and if not, why not, it's lazy if it looks like you've given up, but at the same time don't try too hard because everyone knows you look a bit tragic trying to dress too "young" for your age... And so on, and so on.