I have a good friend who is US diplomatic courier and as Dudemon implies, it sounds exotic but is really not a bed of roses, even if you did get great pay--which you won't. You often do lots of fast-turnarounds, but sometimes extended trips hopping around to a few places before returning to home base. Often the courier doesn't see much more than the cargo/tarmac side of the airport, then out again. Some places you have to travel are darn unsavory, others flatly unsafe where you're not even allowed to stay overnight.
My friend tells me that due to amount of flying they have to do and resultant elite status, he usually gets upgraded to business class though the base ticket is a negotiated-fare economy one NOT biz class. Seems to be pretty common for intra-Asia couriering. This might depend on where one is based though, and what airlines the gov't is working with out of that location. You do get to keep your miles, but frankly, if you want to earn/keep miles and do it somewhat more on your own terms, get a frequent flying job with a regular company or be a consultant instead, not with the US govt doing something like this.
Not sure how "open" to the public this job announcement really is. I was under the impression most of these jobs go to insiders who have already done some form of US government service. Diplomatic couriers usually aren't part of the regular Foreign Service track and don't have to go through quite the same rigorous winnowing process or language requirements.
PS: Seoul is a "minihub" which can get small mail-bag type shipments into PEK by courier (not regular embassy personnel). Big shipments in volume or size go into PEK by a different method/route. But the main Asian hub for US diplomatic couriers is in Bangkok.