How can I conceal peoples' precise locations in ArcGIS Pro?

Group your data geographically, mapping total positive cases by ZIP code, county, state, province, or whatever area. Start with a small unit of area (maybe a city block) and if you think that's too specific, move up to a bigger unit like a ZIP code until you're comfortable with the level of anonymity.

Depending on where you live, HIPAA or another law like it may have something to say on how much anonymization your data needs to go through before it can be released.

I agree with @Dan C. Adding ambiguity via some form of aggregation to your point data is a much better strategy than trying to modify the basemap or other reference layers.

To count points within your administrative boundary of choice, use the Summarize Within tool, using the administrative boundaries as the Input Polygons and your point data as the Input Summary Feature.

You could also try using some form of density calculation to add ambiguity to the data and display clusters with tools like Kernel Density or Calculate Density. The latter may be particularly useful in your case, as you can choose an arbitrary bin shape (square or hexagon) and size to use.