SYMPTOM:
Prior to Strategy Web 9.3.1, users could drill in ESRI maps following a predetermined order of precedence. The three drilling options for the ESRI Map widget are listed below:
When Edit Links Drilling was enabled, it would override the other two drill options, and custom drill paths would override system heirarchy drilling. However, in Strategy Web 9.3.1 and later, even when Edit Links drilling and custom drill paths are disabled. users are not able to drill using system hierarchy for attributes. The drilling icon disappears from the options as soon as the prior two drilling options are disabled, as shown below:
With Edit Links drilling enabled:

With Edit Links drilling disabled:

Notice the drill button has disappeared.
CAUSE:
This is working as designed in Strategy Web 9.3.1. Drilling using the system hierarchy for attributes is no longer supported for ESRI maps.
KB221359