I've never tried arm coolers, they might work well, but in the context of covering up being counterintuitive to cooling down, long clothes typical of Saharan regions were mentioned. It seems to me there are two big differences: Firstly, the clothes typically worn in Sahara, Arabian Peninsula, etc, are all-enveloping but also loose. They allow air to circulate inside them, over your skin. Almost the opposite of cycle clothing! This works to keep cool; I've ridden in India wearing kurta (long, loose cotton shirt worn there) and it definitely keeps you cool, but it's probably not ideal for long distance cycling: very un-aero for a start, and not good if it rains, though the pockets are useful.
The second difference is that desert regions are dry (of course) and a British summer can be hot but even if dry weather, the air is usually humid.
So I'm not saying arm coolers can't work, just questioning if the way in which they work is the same as desert robes.