Most kids would have high school on their linkedin page well past their college graduation, even if it’s just a mention. I’ve seen a lot of linkedin pages. I’ve seen some HS GPAs, and certainly a lot of extra curricular interests like dance etc. That’s who they are.
Some people note national recognition – e.g. Regeneron award, or IMO or things of that kind. The IMO level stuff will go a long way. Even for PhD admissions on the margin. IMO kids are supposed to be more productive in research than non IMO kids, other things equal – someone did a study.
Some of these things get you called for interviews, depending on the industry – e.g. for quant finance. My son suspects that your high school may matter on the margin for getting interview calls for some of these companies. Because he noted that large bunches of kids were from particular high schools in one particular internship cohort – e.g. 5 kids from TJ out of a 100 last summer at his internship. Three-Four from his own HS. Some from Harker on the west coast etc.
I would suspect if a kid is from TJ or Stuy, he/she will mention it, and may get some benefit out of it if he is a val or something like that.