Captcha is not mandatory

You need to handle the submission of the form. For the Contact Form 7 plugin you can follow these directions to get it working.

https://www.jsterup.com/dev/wordpress/plugins/flexible-captcha/documentation/adding-captcha-to-contact-form-7/

Maybe this will help.

The built in forms work without any coding from you but when you have created a custom form there is no way to tell where that form is going to submit to. So you need to handle the submission which you should already be doing with your custom code otherwise your form would be useless.