This sounds like some sort of server cache not clearing. Siteground should be able to find the issue at the server level.
What if you disable the “Cache dynamic CSS” option in Customize > General? I doubt it will work, though, as even the logo isn’t displaying on the main site.
I’ve reached out to SiteGround and expect to hear back from them soon. Yesterday, they cleared the server cache and deactivated it. On the site, “Cache Dynamic CSS” is already disabled.
The issue turned out to be pilot error. One of our new users apparently set the semi-hidden Customizer action setting to Save Draft, instead of Publish. In Draft mode, Customizer changes are not reflected on the live page. Once set on Publish, the Customizer instantly started behaving as expected. See screenshot: