Remove `View post` Text

I think the issue is this piece of code $messages['game'] = $messages['post'];. $messages will contains all the updated messages for all custom post type. In this case, your CPT is game, so you should change the array item of game only.

The correct code should be

public function game_updated_messages( $messages )
{
    $messages['game'][1] = 'Post updated.';
    $messages['game'][6] = 'Post published.';
    
    return $messages;
}

You can try to dump the variable $messages and see all the result it returns.