Although my Apple IIe Platinum was not working correctly, I remembered that the color pattern I was seeing was actually a self-test mode that the Enhanced Apple IIe goes into without a keyboard connected. My keyboard was connected, but was also shattered and seriously damaged. Fortunately, another Apple IIe keyboard saved the day.
Unfortunately, a2usbdsk on the Raspberry Pi is not able to send data fast enough for the Apple IIe to read... yet. I have some code optimizations I need to test out which I think may help the situation. That said, even my old MacBook air was barely able to keep up. My newer 1.8 GHz dual-core Intel Core i5 MacBook Air seems to have no problem. I'll keep trying this weekend.
The Raspberry Pi is able to keep up! The problem is that the FT232H is stalling very often. I had this problem early on in the development of a2usbdsk on Mac OS X, but I worked around it by writing most of my own read/write loops and avoiding libmpsse for the most part. There must be some difference in how libusb or libftdi is handling the packets on the two operating systems. By frequently purging the buffers, I was able to get a disk to load! Need to find a better solution though. More testing should narrow down the problem.