So is it possible to build a simple circuit that does what the FTDI is
In short: it's not possible to build a simple circuit to do that.
You could build a not-so-simple circuit, which would involve either an application-specific integrated circuit, like the FT232 or his cousins (MCP2200, PL2303, CH341, etc), or a microcontroller with a hardware USB-interface integrated. These two are pretty much the same, within an FT232 you would most probably find a microcontroller with a hardware USB-interface. What you gain by using an application-specific IC is that you don't have to write the firmware for the IC yourself. What you gain by using a microcontroller is that you have much more freedom, as you're not bound by the limitations of the hardcoded firmware. Also, given the large offer of microcontrollers on the market, you can probably find a solution with a microcontroller with a lower price tag.
However, in most of the cases, there is usually no sane reason to use a dumb microcontroller for your computing needs (an Attiny45, for example), and put another (more equipped) microcontroller in front of the other to provide the USB interface. You could merge the two, and select a kind of a microcontroller for your needs which also happens to handle the USB interface as well, there are plenty of such microcontrollers.
Of course, sometimes there are legitimate reasons to separate the two functions. Usually a legitimate reason is galvanic separation: putting a pair of optocouplers is easy if you have a pair of serial lines, and much harder if you have a USB line.