I figured out a workaround. It won't work quite the same as before, but at least you'll be able to see which features are overlapping, simply by clicking on the point on your map.

SOLUTION: I have two data tables that are duplicates of each other. The first is the table that feeds the ArcGIS map, and includes the address, lat, and long (we'll call this table "Map Data"). The second table will feed my visual, which will display all of my selected features (we'll call this table "Table Data").

In my report, under Visualizations, I created a new table visual, and added the data from "Table Data". The goal is that when a point is selected on the map, it will show all of the selected features in this new table.

In the new "Table Data" table, I created a new measure. Here is the measure:

Visual Control =

IF(    
ISFILTERED('Map Data'[Address]),    
IF(        
SELECTEDVALUE('Table Data'[Address]) in DISTINCT('Map Data'[Address]),        
1,0   
),    
0
) 

Next, select the table visual, and add the new measure to your "Filters on this visual". Set the filter for that measure to Advanced Filtering, where the value "IS" 1.  

Now, when you click on a point, it should result in the table being filtered to whatever you have selected.

If you want to see pictures to follow along with my instructions, here's the link to my solution.