The AllWinner A13 is basically an AllWinner A10, but with a bunch of integrated peripherals removed, such as HDMI, SATA, I2S, etc.
The only use I see for it is if you have an product that works with the reduced number of peripherals, _and_ can live in the integrated 64MB of RAM. The chip being TQFP is kind of moot if you need to put a BGA-96 DDR chip next to it.
It's certainly doable to do a product with the AW chips. The datasheets are out there, and the mainline kernel (through the sunxi linux efforts) supports a lot of functionality in the chips.
For the more "established" (read: western) manufacturers, I've evaluated TIs Sitara (aka, beaglebone, aka, Octavo), but didn't like their codebase. Quite old and hacky. Also, only single-core, and no upgrade route.
I've settled for the NXP iMX6ULL chips (fairly cheap) for more price sensitive designs, and iMX6Dual/Quad for more powerful designs that require 3D acceleration. It allows me to be a bit more scalable than I would have been with TI, and allows me to re-use my know-how and software base.