The problem is that there are some types of “judging” that cross the line into areas that are prohibited by discrimination laws.
And any person in a management position should be aware of those laws and not simply become a conduit for reinforcing such discrimination.
A professor can choose not to hire a grad student for any number of reasons, but not because the student is Chinese or the professor has overheard the student speaking Chinese. (That’s discrimination based on national origin, and strictly prohibited).
So an email from someone in a leadership position cautioning against doing something which is legally protected because of implications of being judged by such conduct simply isn’t appropriate, ever.
The same would go for suggestions about other things that might impress future prospective employers negatively: wearing a hijab (religious discrimination); a suggestion that a woman should dress more femininely (sex discrimination); a suggestion that an older woman should dye her hair to cover the gray (age discrimination); comments about african-american hairstyles like afros or cornrows (race discrimination); etc.
So part of the responsibility of being the director would be awareness and understanding of anti-discrimination laws and policies. If in fact the professors asked for pictures because they wanted to note down the names of the offending Chinese-speakers in case those same students later submitted a resume to them seeking a research position – then the director should have told them that was improper – and given them a quick lesson or refresher about their obligations to refrain from unlawful discrimination. Instead, this former director apparently reinforced those thoughts by complying with their requests as well as conveying the sentiments to students within the program.
So yes, “others are watching and judging” is a legitimate concern at a university or workplace, but there is a broader area of legal prohibitions against discriminatory conduct that means that some issues are never properly the subject of such cautions.
And yes the lines can probably get fuzzy. Part of the job of being the director would be to know where those lines are drawn.