Finally isolated the issue. We have the Custom Admin Interface plugin installed and use it to alter the menu structure to be in a different order. After some tweaking of that plugin, I was able to add Hubspot back into the menu system, properly this time. So, case can be closed.
I am an admin, I set the site up. Does the plugin require XMLRPC to be enabled? We have that turned off via a security plugin, although briefly disabling the plugin did not fix the issue.
I have been upgraded to Super-User permissions for this client’s account in HubSpot and still have this issue. I tried uninstalling and reinstalling the plugin to no avail.