I have tried below steps in your website:
1) Edit file wp-config.php, add below lines 72~74:
ini_set('log_errors',TRUE);
ini_set('error_reporting', E_ALL);
ini_set('error_log', dirname(__FILE__) . '/error_log.txt');
See our document:
https://toolset.com/documentation/programmer-reference/debugging-sites-built-with-toolset/
section "PHP Debugging"
2) Login as admin user luoyang@cute****.de:
a) Upgrade the "Jetpack by WordPress.com" to the latest version 8.4.1
b) create a new customer user "test-customer"
3) Open a new incognito window, login as user "test-customer":
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4) Open URL:
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click button "Registrieren", fill and submit the form, it works fine, it redirect me to checkout page:
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See screenshot test-customer.JPG
5) Deactivate/Activate the Jetpack plugin again, but this plugin requires to setup the credentials again, since I don't have your credentials, so I have tested step 3) and 4) again, it works fine, it redirect me to checkout page correctly.
Can you setup the Jetpack credentials again? in case the problem still persists.
6) And there are some PHP errors in your website, you can get them in the website root folder, file "error_log.txt", for example:
WordPress database error Table 'db823585745.EVzgcaJiblc_links' doesn't exist for query
SELECT COUNT(*)
FROM (
SELECT 0
FROM
EVzgcaJiblc_links AS links
JOIN EVzgcaJiblc_instances AS instances ON links.link_id = instances.link_id
WHERE
( instances.parser_type IN ('link', 'image', 'youtube-iframe', 'metadata', 'url_field', 'acf') ) AND
( instances.container_type IN ('comment', 'post', 'page', 'dummy') ) AND
( ( broken = 1 ) ) AND
( (dismissed = 0) )
GROUP BY links.link_id) AS foo made by require_once('wp-admin/admin.php'), require('wp-admin/menu.php'), require_once('wp-admin/includes/menu.php'), do_action('admin_menu'), WP_Hook->do_action, WP_Hook->apply_filters, wsBrokenLinkChecker->admin_menu, blcLinkQuery->get_filter_links, blcLinkQuery->get_links
The table ***_links is WordPress built-in database table, see document:
https://codex.wordpress.org/Database_Description#Table:_wp_links
And I don't find it in your database dump file:
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You might need to check them in your website databases
7) And you are using outdated Mysql server:
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section "Outdated SQL server", I suggest you try to upgrade the mysql server to 5.6 or above.