After discussing with a few other people, the purpose of providing an extent, height, width and dpi is so that the map service can determine the current map scale. Once the scale is known, the map service can exclude layers based on their scale dependency settings. The map service is not doing a spatial intersection based on the extent that's provided to exclude layers from the identify operation.
Glad you were able to come up with a solution to get your identify time down to something relatively reasonable. For future reference, when you have hundreds of layers you need to serve, the Esri recommendation is to use multiple map services with the number of layers in a map service to something smaller, say...less than 100. Preferably less than 50.
Could that explanation be included in the Class: IdentifyParameters API Reference? What is currently in the documentation is not clear on that point (hence this thread).