I am having the same issue. I am using just first name and email, and set the field ID names in Elementor to FNAME and EMAIL to match the mailchimp field tag values shown when looking at the mailchimp form builder tool.
Regarding "Actions after submit", If I turn off the mailchimp action in Elementor, both the email action and redirect action work fine and I end up at my thank you page. As soon as I enable mailchimp again, nothing works and I do NOT get the message saying the form was submitted successfully, nor does the redirect take place. On a samsung smartphone I get a Red X after trying to submit, but on a windows 10 machine with Chrome, no error, no red X.
Is there any Wordpress.org or Elementor logging left behind on my site somehwere or from my WP admin console??
If I also go into Integration settings under Elementor->Settings and then the Integrations tab and specify my Mailchimp API key, and then click validate API key, I so end up with the green check mark, so it appears to work.
On the Mailchimp side, when checking Integrations, I do not see anything for Elementor, but there is a WordPress integration. Should I go through that process? It generates a small piece of code, and suggests that needs to be pasted into the WordPress theme header.php file.
Any thoughts on how to further troubleshoot this? Thank you.
I am using the free versions of mailchimp and the Elmentor plug in for Wordpress if that matters.