What’s the downside of a login session?

Using the following simplified login form:

// login_form.php
<?php session_start(); ?>
<?php if ($_SESSION['isLoggedIn'] === false) : ?>
   <form action="login.php" method="post">
      <input type="password" name="password" />
      <input type="submit" value="Login" />
   </form>
<?php else: ?>
   You're logged in!
<?php endif; ?>

// login.php
<?php
session_start();
$_SESSION['isLoggedIn'] = htmlspecialchars($_POST['password']) == "my secret";
header("Location: login_form.php");

I want to understand, if there is a downside using this “type” of login and I want to understand why the following is not working:

• Navigate to login_form.php with a cleared session
• Open devtools and create a session isLoggedIn=true manually
• Refresh the page
• Result is not logged in