The first thing to try is deleting all files in the core/cache folder and visiting the site with your browser in private or incognito mode, but it sounds like you may not have that option.
Repairing all tables in the DB is another thing to try.
In my experience, about 90% of 500 errors are caused by an issue in .htaccess, though that typically doesn't affect the Manager.
Another possibility is an older plugin that's causing trouble. If you have access to the DB, you can disable all plugins in the modx_site_plugins table (put a 1 in the 'disabled' field).
If the core/cache directory is truly missing (and I can't imagine how that could happen), MODX would try to create it and possibly crash with a 500 error when that failed.
In light of that, you might want to just ask the host to create an empty core/cache directory and make sure it's writable by PHP.
I applaud you for leaving. I would never host with a company that didn't provide cPanel access (or a
host owned by EIG). For future reference:
https://bobsguides.com/modx-friendly-hosts.html