To provide GTM with metrics to select the best datacenter -- but the GTM wants a "load object" which is just a file that reports the current load on a datacenter (I suppose it doesn't strictly have to be HTTP/S). And the GTM doesn't have access to poll each individual webserver, it only has access to the BigIP LTM we have sitting in front of the webservers.
So we can get the LTM to provide this "load object" on request, with some arbitrary load value, but that wouldn't be entirely useful. We'd like to calculate the load value based on page load times of webservers but we can't figure out a way to do this without, as you said, a lot of string and duct tape. Maybe the BigIP LTM isn't suited for this kind of thing -- are we better off doing all this on some other server and just using the LTM to pass requests for the "load object" to that server?
Nimbostratus