More reason for me to distrust public school systems. Big red flag in an educator if they're trying to "get to the kids" early about anything other than what the agreed upon curriculum is.
If the chess club coach notices that a student displays aptitude at strategic thinking, would it be fair to invite them to attend chess club? If the student's parent is upset by the child attending chess club, does the school have a duty to inform them, or does the child have freedom of association?
That said, I'm pretty uncomfortable with teachers having visibility into kids' Google searches, unless the kids know that they're under such scrutiny.
Put another way, if a teacher isn't teaching the bible, clearly they are agents of the devil right?
It's the same poisonous fallacious logic that equivocates a policy position with basic moral character.
CRT theorists propose the same binary: those who are not pro-CRT support racism.
It is possible to be against both racism and CRT.
CRT is not scientific. It is ahistorical, based on storytelling and disagrees with basic fundamental statistics. It is a religion with followers and evangelists, demons and devils. It even has a salvation arc, original sin and the "sins of the father" as concepts. Blood guilt is not science.
Having an opinion is not a crime neither is being straight, gay or whatever you can think of as long as it is safe, sane and consensual.
And generally, when a kid shows strategic thinking, the school tells the parents about it and the chess club. There may be reasons why parents don't want their kid to play chess, and if it is not part of the curriculum, it is not something for the school to decide, they can make suggestions but parents have the final word.
Also, noticing strategic thinking does not need to involve spying. If the chess coach looked as search histories for recruitment, that would be a problem.
There's orders of magnitude difference between the chess club coach noticing a student displays aptitude and inviting them to join the club and a teacher _actively searching_ a students computer for "evidence" of them supporting the teachers politics-du-jour.