Please help me get the php appointment form work

I am not very experiencing with php and I need help. Basically I am trying to create an appointment field so that when people fill it I get an email with the input people put.

So in the first website which is in a file called appointment.hmtl I have the following code:

    <section class="section appoinment">
 <div class="container">
    <div class="row">
      <div class="col-lg-4">
          <div class="mt-3">
            <div class="feature-icon mb-3">
              <i class="icofont-support text-lg"></i>
            </div>
             <span class="h3">Need more information? Please give us a call.</span>
              <h2 class="text-color mt-3">+954-225-4454 </h2>
          </div>
      </div>
            
            <div class="col-lg-6 col-md-10 ">
                <div class="appoinment-wrap mt-5 mt-lg-0">
                    <h2 class="mb-2 title-color">Book an appoinment</h2>
                    <p class="mb-4">Just fill out this simple form and your appointment will be reserved.</p>
                     <form id="#" class="appoinment-form" method="post" action="mail.php">
                    <div class="row">
                         <div class="col-lg-6">
                            <div class="form-group">
                                <select class="form-control" id="exampleFormControlSelect1">
                                  <option>Preferred method of contact</option>
                                  <option>Call me</option>
                                  <option>Text me (fees may apply)</option>
                                  <option>Email me</option>
                                </select>
                            </div>
                        </div>
                        

                         <div class="col-lg-6">
                            <div class="form-group">
                                <input name="date" id="date" type="text" class="form-control" placeholder="dd/mm/yyyy">
                            </div>
                        </div>

                        <div class="col-lg-6">
                            <div class="form-group">
                                <input name="time" id="time" type="text" class="form-control" placeholder="Time">
                            </div>
                        </div>
                         <div class="col-lg-6">
                            <div class="form-group">
                                <input name="name" id="name" type="text" class="form-control" placeholder="Full Name">
                            </div>
                        </div>

                        <div class="col-lg-6">
                            <div class="form-group">
                                <input name="phone" id="phone" type="Number" class="form-control" placeholder="Phone Number">
                            </div>
                        </div>
                        
                        <div class="col-lg-6">
                            <div class="form-group">
                                <input name="mail" id="mail" type="text" class="form-control" placeholder="Email">
                        </div>
                </div>
                
                    </div>
                    <div class="form-group-2 mb-4">
                        <textarea name="message" id="message" class="form-control" rows="6" placeholder="Your Message"></textarea>
                    </div>

                    <a class="btn btn-main btn-round-full" href="Eng-confirmation.html">Make Appoinment<i class="icofont-simple-right ml-2"></i></a>
                </form>
            </div>
        </div>
      </div>
    </div>
  </div>
</section>

And in another file called mail.php I have the following code:

    <?php

    if ($_SERVER["REQUEST_METHOD"] == "POST") {

        # FIX: Replace this email with recipient email
        $mail_to = "example@gmail.com";
        
        # Sender Data
        $subject = trim($_POST["subject"]);
        $sphone = trim($_POST["phone"]);
        $name = str_replace(array("\r","\n"),array(" "," ") , strip_tags(trim($_POST["name"])));
        $email = filter_var(trim($_POST["email"]), FILTER_SANITIZE_EMAIL);
        $message = trim($_POST["message"]);
        
        if ( empty($name) OR !filter_var($email, FILTER_VALIDATE_EMAIL) OR empty($subject) OR empty($message)) {
            # Set a 400 (bad request) response code and exit.
            http_response_code(400);
            echo "Please complete the form and try again.";
            exit;
        }
        
        # Mail Content
        $content = "Name: $name\n";
        $content = "Topic: $subject\n";
        $content = "Phone: $phone\n";
        $content .= "Email: $email\n\n";
        $content .= "Message:\n$message\n";

        # email headers.
        $headers = "From: $name <$email>";

        # Send the email.
        $success = mail($mail_to,$content, $headers);
        if ($success) {
            # Set a 200 (okay) response code.
            http_response_code(200);
            echo "Thank You! Your message has been sent.";
        } else {
            # Set a 500 (internal server error) response code.
            http_response_code(500);
            echo "Oops! Something went wrong, we couldn't send your message.";
        }

    } else {
        # Not a POST request, set a 403 (forbidden) response code.
        http_response_code(403);
        echo "There was a problem with your submission, please try again.";
    }

?>

However, when I test the website and fill the field I never receive the mail with the information. This code is taken from a free template website so I know is kind of right, but there is something else I a missing or need to add for it to work.

Thousands thanks to the kind soul that help me make this work.