I think I've at least found the location of the problem:
core\model\modx\processors\system\config_check.inc.php
MODX seems to use that file's existence as an indicator of whether the core has been moved. The cURL response is probably not the one MODX is expecting. I had a similar problem with SiteCheck.
I thought there was a way to turn off the config check in System Settings, but I couldn't find it.
If I'm right about what the problem is, there are several possible fixes.
1. Edit the default Dashboard (System -> Dashboards) and remove the Configuration Check widget.
2. Rename that readme.txt file. (I think this would work, but it would be undone by every MODX upgrade).
3. Move the core above the web root. This would be my choice. I feel safer with the whole core absolutely inaccessible from the web and in a location few hackers would look for it. It's not difficult to do. You only have to do it once. UpgradeMODX will still work. And you can create a Media Source that lets you still see and edit the core files on the Files tab in the Manager.