I hope you are well today and thank you for your question.
Please see the information provided in the following topic for the similar question.
If you still have any question then please ask it in this topic.
You don’t have to configure in the Writing setting as this setting is for posting to WordPress by email and not for making emails work on the site.
Are you getting other emails from your site like user registration or comments?
If you find it’s hard to configure SMTP settings then instead use WP-mail send method in the contact form settings.
Im not getting any emails form the site whatsoever.
Then it is not contact form problem but your site problem that you need to troubleshoot and resolve.
You can troubleshoot the email issue as described on the following pages or contact your web host regarding this.