Using multiple polygons in shapefile to find intersections

The goal of this workspace is to go through multiple shapefiles and update a field for each polygon in a shapefile. The field will be compiled of the sum of an attribute from another shapefile that intersects with a single polygon in the shapefile.

 

Ex. I have a 15 minute drive time polygon from a single point (a single polygon in a shapefile of many polygons). I want to see how many block groups intersect/are within the drive time. I then want to sum or average a field within the block groups and then add this new number as a field to the drive time shapefile. Then it will move onto the next polygon, maybe a 30 minute drive time, and do the same.

I have been attempting to use the Spatial Filter but I am unsure if this creates intersections for each polygon or if it is assessing if the candidate needs to intersect with all polygons read from the shapefile.

 

I am open to using arcpy as well to help solve this problem. Overall I think I am asking if looping through each polygon in a shapefile is possible and then updating by row.