I'm concerned about the TCP stacks on these boards. Near as I can tell, they're all built into the chips and no source code is available. Is my impression correct?
Yeah, they're pretty much black boxes. You are literally talking to them like you used to work with modems. "AT+" commands and all of that.
You can program the ESP8266 yourself. For its TCP stack it uses lwip which you can also compile yourself. The 802.11 stack is more of a black box though, but not necessarily the TCP stack.