try_this.php

# instead of testing for the user existing, test if the user doesn't exist
function user_not_exists($username)
{
    $connection = $this->get_connection();
    $stmt = $connection->prepare("SELECT `username` FROM users WHERE username = ?;");
    $stmt->bindValue(1, $username, PDO::PARAM_STR);
    $stmt->execute();
    return empty($stmt->fetchAll(PDO::FETCH_ASSOC));
    # PDOStatement::fetchAll returns an array, so it's easiest to test if the array is empty (no results found)
    # if you want to keep the function as user_exists(...), try returning $stmt->fetchAll(PDO::FETCH_ASSOC)[0]
    # so that you get the actual result, and not just the array 
}