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I don't know what they are smoking over there at SiteGround, but Akeeba Backup has absolutely nothing to do with caching headers in the frontend of your site. Akeeba Backup is a site backup software.

If they meant Admin Tools, ditto. We do not touch the caching headers for Joomla–generated content. Quite the contrary, our .htaccess Maker allows you to enable caching for static media files by setting their expiration time into the future. That is to say, it will help your server serve the site more efficiently and make your visitors' browsers perform fewer requests to your site.

Your issue with the dynamically generated pages, however, is unrelated to any extension.

Joomla itself always sends Cache-Control headers (also in \Joomla\Application\AbstractWebApplication::respond which is part of the Joomla! Framework) to bust the cache of any page unless you enable Joomla's cache in Global Configuration and enable the “System - Page Cache” plugin. This has been the case since Joomla 1.5 — not a typo, one point five, released sixteen years ago.

Of course is you set Joomla's Cache to Conservative and enable the “System - Page Cache” plugin you will indeed be allowing Joomla's dynamically generated pages to be cached. In this configuration SuperCache will be able to cache the pages just fine.

Please note that using their SuperCache (which is nothing more than Varnish HTTP Cache in front of the web server) you will get exactly what you are asking for: all public pages will essentially perform as if they are static pages, their contents keyed to the URL. Pagination may not work on all extensions, definitely not if they are using the user state to determine which page of information to display without modifying the URL at all. Dynamic content for guest users depending on external attributes (e.g. showing different content to guests depending on whether they have accepted cookies, without using JavaScript) will not work. Any form on your page which uses an anti-CSRF token including Joomla's login module itself will not work (they will fail with an “invalid token” error). Modules or plugins which pull dynamic content at the server side, before rendering a page, such as random articles, random images etc will not work. This kind of setup only makes sense when you have a fairly static public site such as a news site.