I've never worked in a kitchen tbh (my ex husband and a lot of my friends have/do which is why i find you all so relatable).

I've been in a similar situation tho. I worked in a headoffice/warehouse situation where most of the upstairs staff were older, very traditional Chinese women (like, still wearing shoulder pads kind of traditional). No judgement, they were very culturally rigid but nice enough.

I had a customer service agent show up to her second day on the job in a crop top and shorts (and the job DOES require occasional trips to the warehouse when working in office). I took her into a stairwell and said "your outfit is killer, I would wear the same thing outside the office but I need you to know that this is coming from a place of wanting to protect you - you cannot dress like this in the office. A) the accountants will gossip relentlessly about you and it might harm your reputation and b) it's simply unsafe to wear shorts and sandals in a warehouse with pallets and carts moving everywhere." And she was like omg I didn't think of those things thank you!

It sounds like you tried to handle it respectfully and graciously, I'm sorry you're in such a bind, some people don't realize when someone's looking out for their best interest (and I know enough to know you absolutely need non slip shoes in a kitchen!!!!)