This code snippet ensures that the CF7 form is always initialized in popups created with Elementor. If Turnstile is implemented, it will also work correctly.

wp_fix_elementor_popup_cf7_forms.php

add_action('wp_footer', 'fix_elementor_popup');

function fix_elementor_popup() {
    ?>
    <script>        
		jQuery(window).load(function () {
			jQuery.each(elementorFrontend.documentsManager.documents, (id, elementorDoc) => {
				if (elementorDoc.getModal) {
					elementorDoc.getModal().on('show', () => {
						// REQUIRED
						// Re-initialize Contact Form 7 when the modal is shown
						jQuery(".wpcf7 > form").each(function () {
							wpcf7.init(this);
						});

						// OPTIONAL
						// Check if turnstile is defined before trying to reset or render it
						if (typeof turnstile === 'undefined') {
							console.warn("turnstile is not defined!");
							return;
						}

						// REQUIRED if you have Turnstile on the page, you need to reset or render it when the modal is shown
						// Reset or render Turnstile for each element with the class 'cf-turnstile'
						// Implementation with https://br.wordpress.org/plugins/simple-cloudflare-turnstile/
						jQuery('.cf-turnstile').each(function (i, el) {
							var widgetId = el.dataset.cfTurnstileWidgetId;
							if (widgetId) {
								console.log('reseting current turnstile widget ', widgetId);
								turnstile.reset(widgetId);
							} else {
								el.innerHTML = '';
								turnstile.render(el, {
									sitekey: el.dataset.sitekey,
									callback: window[el.dataset.callback] || function () { },
									theme: el.dataset.theme,
									language: el.dataset.language,
									size: el.dataset.size,
									appearance: el.dataset.appearance,
								});
							}
						});
					});
				} else {
					console.warn('elementorDoc.getModal is not defined!')
				}
			});
		});
    </script>
    <?php
}