I have recently set up a WordPress Archive page for one of my taxonomies 'Countries'.
While these pages seem to load relatively okay for guest users, some of those same pages simply won't load whilst I'm logged in as Admin.
For an example, log in using the credentials provided and head to hidden link
If the page does load, that's because it's been cached - To clear the cache for the page go to 'WP Rocket' in the top bard menu (beside 'Front-End Layouts Editor' button) and select 'Purge This URL'. The page will then refresh and will likely load very slowly (if at all).
I've never has such a slow speed issue with my other large archives on the site so any assistance would be most appreciated!
For reference, I recently updated all Toolset plugins to their latest versions and the php from 5.6 to 7, but this issue was present both before and after these updates.
I notice you have Query Monitor installed, and I activated that but it doesn't help because the page doesn't finish loading and so I cannot look at its results to spot what is causing the issue.
I'm taking a copy of this updated site to install locally so that I can take a closer look.
Looking at your debug info your server settings seem more than adequate.
I installed the copy of the site, and I could see the same problem, with it taking a very long time to load the country archive page.
I disabled the performance plugin WP Rocket, and that seemed to be the cause. The page loads, not blazingly fast, but in just over 5 seconds.
One time Query Monitor reported a slow query, a relationships query. Looking through the custom archive, and the templates and Views used within it, the only thing I noticed was that the View "Single Item Card - Voyages Available Banner (Voyage as Child)" contains a relationship Query Filter, which specifies "any" relationship. If you know which relationship is pertinent here you should specify it, which should help a little with the query time, but will in any case only have a minor effect on the actual page load time.
When I reactivated the WP Rocket plugin it ground the archive page to a halt.
I'm not too familiar with it, I don't know if you can exclude it from the archive page in its settings...