Was it not possible for the police to force a logout from the backend? Or at the very least, update the game to where a logout is possible?
No, it was not possible for the police to force a logout from the backend. The players were connected to the game using Nervegear. Nervegear connected to the player using microwaves, allowing the player to be “attached to the game.” Their mind was the controller. Imagine a corded Xbox controller that self-destructs when unplugged... the internal battery in the gear prevented the player from disconnecting without being killed. Even if the servers were blown up, the headset would kill the players.
And no, there was no way to update the game to make logging out possible. In Episode 12, "Yui’s Heart", it's revealed that the game is self-contained and would not have allowed an outside update.
If the game had been designed to be accessible from the outside, then an update to allow the players to log out would have been possible, but it was designed so the players could not log out. There was even a spot for the logout on the menu, but it was removed and the players were stuck.