Security through "obscurity" in this context means using anything but a known and common standard. It's easy to show many examples of this type of security gone wrong. But that's not entirely due to poor decisions on the part of the implementers, it's also due to the prevalence of this kind of security. Looking at the failures alone is educational, but ultimately misleading as to the effectiveness of this kind of security. One must also consider that for every one that went wrong, many more went right.
That could even include some systems that are technically weak, and easily broken. I have found security exploits on a few devices. One is quite scary in terms of what it could be used for. But this was done for personal use only, to get around some limitation of the device. I didn't use my knowledge write a virus. Nor did I publicly document it like some security "researchers" tend to do, under a false flag of increasing security awareness. And no one else appears to have done this either. Therefore, the public can continue using these devices, without fear of script kiddies like Mallory walking around and causing mischief. Even if something can be exploited, there is no issue unless the exploit actually enters the wrong hands.
As for the Allwinner A13 being $2, that is very likely a bogus claim, originating from a particular Kickstarter for "CHIP - The World's First Nine Dollar Computer". Many people have analyzed the BOM for this and found $9 to be completely unrealistic, concluding that it's either a scam, or a loss-leader on which they're recovering the loss based on sales of overpriced add-on boards. In particular, Olimex concluded this product actually has a BOM of about $20. Part of that was getting a quote from Allwinner on the A13, which was $4.80 in quantity 5,000.
Furthermore, given the complexity of the A13, versus the quality of support and documentation from Allwinner, rolling-your-own product around the bare IC would likely be a painful process; significantly increasing development cost. The A13 recommendation is not realistic IMO.
Finally, I'm curious. As it was not stated in this thread, does anyone here actually know what [nctnico]'s device does? If not, then recommendations for Linux and idealized security might prove to be a bit silly, if it turns out he's developing an Ethernet-controlled dancing Coke can.

(Or something equally innocuous if hacked.)