Do you have any reporting corollaries for other resources you are providing? What measures are acceptable to your upper-level administrators for those? Can you measure the same parameters for GIS? If so, then even if they aren't familiar with the platform, then at least they are familiar with the measures.
As you note, raw numbers, like the total number of users, are not always a good way to quantify resource use. A quick and easy improvement is to look at usage numbers over time, such as how many unique users use the resource each day, each week, each semester? In other words, what is the distribution of the resource's user base across people using it once to those that depend on it regularly? (This kind of data carries sufficient weight with our institutional stakeholders, rather than the raw numbers.)
This is, of course, an under-representation of GIS use at an institution. It is strictly tied to Esri ArcGIS products, and even for those fails to account for use via Single-Use or Concurrent-Use licensing. (If you have access to your Concurrent-Use logs, then you can integrate that data.)
You also likely have plenty of users using R, QGIS, Google Earth Engine, Carto, Mapbox, and other non-Esri products to do GIS. Some of those can also track and report on usage, but it gets complicated fast.
Another set of quantitative measures we track for reporting center around supporting the users of GIS, and those data are GIS platform agnostic. From our ticketing and work-planning system, we can produce reports on how many GIS support requests we receive, where they come from, and how much effort it takes to resolve them. We can also look at the life-cycle of the requests; for instance, if it is taking an unreasonably long time to assist users, then we can use the data to justify to administrators the need for additional GIS support resources.
Hope that helps!