The game tells you to delete the logs on servers because computers can track all incoming requests, which can be a problem when you are doing nefarious actions and looking through confidentional infornation. Tracking who is accessing a computer/server and why has been a common practice, especially for buisnesses.
You also don't actually need to delete the logs while playing. There are only 1 or 2 minor moments where there is a consequence for not doing that, and in those cases a fork bomb is also needed (to prevent leaving a disconnect log).
The actual scary part of the game is (SPOILERS) a big tech company developing a massive 0-day exploit and wanting to monopolize the patch for it.
The Last of the Last of Us.