First, check the reset pin info in the data sheet. I'm fairly sure it can be left floating or tied to VccIO. You have tied it to +5 but your VccIO is 3.3V so not sure if that will have done any damage.
Also post your layout. The USB side is not super critical (I've never had an issue with FT232BL/RL) but you still need to observe basic rules for routing and decoupling.
Interesting - unless I missed something the data sheet says tied to Vcc or floating, but then all the diagrams show it tied to VccIO. I tried cutting the track to leave it floating but that didn't help. Hopefully I didn't damage the IC - a quick look on Google suggests it's 5v tolerant on all pins.
You should make sure you're not suffering from serial dyslexia as well. Tx and Rx always have a perspective. WHO is Txing and WHO is Rxing.
The FT230XS chip is labeled from its own perspective. Tx on the data sheet means the FT230XS chip is Txing on that pin, and Rx means the FT230XS chip is Rxing on that pin. You need to tie the FT230XS Tx to your micro's Rx and vice versa. I always find that arrows are a much clearer way of communicating the direction of comms versus the ambiguous "Tx" and "Rx" labels that everybody uses.
This could be at least part of the problem. The board is layed out from the IC perspective, but I could have easily hooked it up from the micro's perspective. Pretty sure I tried swapping the data lines while troubleshooting at some point so it wouldn't be the only problem.
I like the idea of adding arrows though.
Stupid question, but do you have the right driver installed on your PC?
Your PCB layout could also be helpful to troubleshoot the problem, even though it does only full speed USB so it's not critical.
I tried that, didn't help. I'm sure if it was a driver issue the PC would still detect something on the USB port right? I wasn't even getting an unknown device!
I'm attaching the board layout. I colour coded the pads for the caps as they're not a standard footprint - I'm using through hole caps soldered to smd pads.