I would go about this in a different way.
I would add Configurable Reports to do this, by adding it to a course set up for just for those support staff users.
I would create the report they need there, available just in that one course, and then in CR give permissions to their role (non-editing teacher or even student) on the Permissions tab.
This skips the entire issue of having to adjust roles capabilities and tinker with Browse list of users.
In addition, it would allow you to add ANY sql report you could build and the support people would be able to see the data without any possibility they could modify anything: it's just reports.
I have done it this way in the past, where I code the reports, and staff use them to see what it going on in the site, without needing to give them any Moodle permissions to Browse list or Site admin reports at all.
You can even do the requested report - user names and last access times - in CR without using any SQL code at it.