You know the fun thing is, something like the Allwinner A133 - which is one of the most popular SOCs in lower-end tablets today - is like $5, or $3 in quantity.

It turns out it's actually not as hard as you'd expect to whip together your own board with one of those + LPDDR4 RAM + eMMC storage + fixings, and get yourself something like what you're talking about for... I dunno, sub $50? Maybe even sub $20 depending on how much RAM you put on it and what other capabilities you give it.

I'm in the middle of designing just such a board right now. Totally recommend taking a stab at it if you have any EE chops at all (or want to learn!)