It is unclear what brands of passport merit E-Gate cards for non-residents; presumably those that are permitted Visa On Arrival.
At the top of the stairs upon arrival in Dubai Terminal 1 passport hall, there's a small desk on the right. Other Terminals doubtless have different arrangement. YMMV. The Man In White confirmed that the Goat could indeed obtain an E-Gate card, but would first have to arrive and have his passport stamped at the desk around the corner (on yet another fresh page, as usual). The queue at this desk, which is adjacent to the E-Gate terminals, was mercifully short, unlike the rest of the arrivals hall that was standing room only and breathing by numbers.
With passport stamped, the Goat was directed over to the National Bank of Dubai booth, there to hand over Dh220 and get a receipt.
Now back to the E-Gate desk, where the Man In White took a digital mugshot and scans of the Goat's hoofprints. In exchange for the NBD receipt, the Goat received his E-Gate card.
Easy peasy, lemming squeezy.