Thanks for the tip. My experience this morning confirms this. At the very far left of T3 arrivals, after customs, opposite end to chauffeur drive. Inside the immigration office it is counter number 10. Took about 20 minutes including waiting for an Italian couple to both finish their registration. There's a machine at a counter by the entrance which issues numbered tickets, and clients are seen in turn. There are several buttons on this machine including one for "EGATE". Although a big flatscreen LCD showed my number being eleventh in line, actually I was seen next - maybe someone put in ten requests in error.

To get back into the area for chauffeur drive/Dubai Connect hotel it was necessary to wait outside a glass one-way auto door for someone to come out...