Intel has parts that would work
(albeit a bit light on GPIO); I've got a dreadful little tablet here based on the Z3735G
, and they packed that CPU, a gig of RAM, 16GB of flash, a 1024x600 touchscreen, some sort of BT and wifi, and a battery together for under $50.
If they hadn't also burdened the device with some of the more agonizing firmware I've had the pleasure of dealing with(AMI's dysfunctional take on 32-bit UEFI, no compatibility support module, on a 64-bit platform? Sign me up!); it'd actually be a decent little Linux toy, since that Atom is supported by intel GPU drivers, not the freaky PowerVR stuff.
As best I can tell, though, the Z-series Atoms didn't attract all that much interest(they are comparable to ARM devices aimed as similar price performance niches; but not particularly superior); and vendors weren't exactly clamoring to buy the chips; and Intel doesn't really like selling parts that cheap. They'd much rather try to sell you on a Core M or the like.
There isn't a whole lot of reason to do it; or apparent interest; but it could be done.