I think it makes sense for people to have their status show as "offline" if they wish to not have people be able to know exactly when they are online or not. Or there could be a setting, "Only show online status to friends".
As for hiding game history - that seems to be a feature of this site, that people can review games from different perspectives to study, learn, and improve.
Making it possible to hide one's history might be a good feature to grant to premium members. I also think that if a person hides his or her game history, it should also hide them from being displayed on the rankings leaderboard. that would encourage some of the top players to leave their game history visible.
As for the actual stalking being discussed. Well, stalking is serious, and should not be permitted. Also, creating multiple accounts is against the rules here, right?
I am not sure exactly what the nature is of the stalking described by the OP though. The explanation was indeed very vague, and it is unclear to me exactly what the alleged stalked is doing.
I also have no idea how anyone knows when someone with a new account is a person who has created a second account to get around the red-thumbs system. I have played games against players new to the site. I have no idea whether the person is actually new, or simply made a new account.