The suggestion to disable caching whether by using a plugin or a built-in feature by the web host is just to test if it's the cause of the issue. The use of caching is recommended in general and in fact, it's mentioned in our performance guide. See https://theme.co/apex/forum/t/customizations-performance/210
Because there are different caching systems though, we can't take everything into account and we do not know what other features a caching system includes. With that said, we can't really explain why the Pro icons disappear when you enable your host's caching system. What we can tell is Pro interface icons are made with SVG. It is the web host who can explain why SVGs could not be displayed in their particular system.
Since you said your other sites are working using the same caching system, there could be something unique in your particular site (oceanbottega.ca) that is causing issue in the first. It could be another third-party plugin, difference in WordPress setup, etc.. You will need to compare everything in your working site and the site with issue.
We are not sure if enabling Super Cacher and HTTPS Enforce will cause issues again. You will need to try activating theme and monitor if the issue will come back.
If it does, please first try purging all caches in Super Cacher and also clear your browser's cache and also test in incognito mode and other browsers.
I'm not sure how HTTPS Enforce will affect the icons since it's only a redirect of HTTP to HTTPS. It could be that the problem is with your WordPress setup. We will need to know first what really is causing the issue.