form-submissions-cpt.php

<?php
/**
 * Intercept form submissions and create posts in a Custom Post Type.
 * 
 * - Intercepts the form-submissions
 * - Registers the Custom Post Type
 * - Adds columns to the admin page for the CPT
 * - Prints data in the CPT columns in the dashboard.
 */

/**
 * Intercept the form submission.
 */
add_filter( 'gutenberg_form_submit_short_circuit', function( $override, $params ) {
	$post = wp_insert_post( [
			'post_type'  => 'form_submissions',
			'meta_input' => [
				'name'    => $params['name'],
				'email'   => $params['email'],
				'comment' => $params['comment'],
			]
		]
	);

	wp_safe_redirect( get_site_url( null, $params['_wp_http_referer'] ) );
	exit;
}, 10, 2 );

/**
 * Add the custom post type.
 */
add_action( 'init', function() {
	register_post_type( 'form_submissions', [
		'label'                => __( 'Form Submissions', 'forms-cpt' ),
		'public'               => true,
		'show_in_rest'         => false,
		'hierarchical'         => false,
		'excluded_from_search' => true,
		'publicly_queryable'   => false,
		'show_ui'              => true,
		'show_in_menu'         => true,
		'show_in_admin_bar'    => false,
		'icon'				   => 'dashicons-feedback',
	] );
} );

/**
 * Add colums to the CPT admin page.
 */
add_filter( 'manage_form_submissions_posts_columns', function( $columns ) {
	unset( $columns['title'] );
	$columns['name']    = __( 'Name', 'forms-cpt' );
	$columns['email']   = __( 'Email', 'forms-cpt' );
	$columns['comment'] = __( 'Comment', 'forms-cpt' );

	return $columns;
} );

/**
 * Print the content of the columns.
 */
add_action( 'manage_posts_custom_column', function( $column_name, $post_id ) {
	$post = get_post( $post_id );

	// Bail early if not a form submission.
	if ( 'form_submissions' !== $post->post_type ) {
		return;
	}

	switch ( $column_name ) {
		case 'name':
			echo esc_html( get_post_meta( $post_id, 'name', true ) );
			break;

		case 'email':
			$email = esc_html( get_post_meta( $post_id, 'email', true ) );
			echo '<a href="mailto:' . $email . '">' . $email . '</a>';
			break;

		case 'comment':
			echo esc_html( get_post_meta( $post_id, 'comment', true ) );
			break;
	}
}, 10, 2 );

The example on form-submissions-cpt.php can be seen in action in the screencast below:

Screen.Recording.2023-06-06.at.10.58.27.AM.mov

It assumes a default form with name, email and comment fields.
If the form contains different fields, the code can be adjusted accordingly.

Note: The above is just a proof of concept to show how easy it would be to use the forms/inputs blocks to implement any kind of behavior, from creating a post to sending requests to 3rd-party APIs and anything in between.
As a proof-of-concept, it doesn't handle security etc - but that would be easy to do by adding some additional checks for the nonce that form submissions include by default.

Prerequisites for the forms to work while this is still an unmerged PR:

1. Checkout the PR on WordPress/gutenberg#44214 and run npm run build in Gutenberg
2. Go from the WP Dashboard to Gutenberg -> Experiments, and enable the Form and input blocks experiment.