How to keep WP from using https to get to wordpress.org?

You don’t need to turn off SSL on the server. You need to install an HTTP client method that supports SSL transport.

This has nothing to do with your server supporting SSL as a server. In this case, WordPress is acting like a web-browser, and trying to make an HTTPS connection to WordPress.org, so that you can download files securely.

Typically, you need to reconfigure your PHP installation on the server so as to have CURL and OpenSSL available to PHP. How to do this precisely depends on your hosting service or system.