i work for FedEx, and they do offer a special ultra high priority service that involves a courier flying first class on the first available flight (any airline possibly even a charter). I was told by my hiring manager (she was explaining the number of different positions i could potentially get into if I stay with FedEx long enough) that there are even the handcuffed briefcases. This is of course an elite service that is not publicly offered on their website for average people (because it costs thousands of dollars for just a small package).
As for the realty of the trope, I've known a number of couriers (both document couriers and precious gem couriers) and I've never heard of any of them using a handcuffed briefcase for transport. Aside from being patently obvious, it would also be very awkward, especially for commercial air transport or in any other situation in which they may need to remove a jacket. In such cases, discretion and awareness are better security than any physical safeguards. Most secure documents are actually either shipped via post or FedEx in special inside packaging but with just a standard external box or envelope, which is frankly probably more reliable than an individual courier for all but the most sensitive information.
I'm wondering if more often than not the handcuffed briefcase a dummy. If you are in a country where the Heads of State are under threat of assassination or whatever, and you see the huge armed police escort followed by the Presidential Limo, and then another trailing armed escort, you can bet on one of two things.