Solution: Tying A10 to GND successfully makes it enter DFU bootloader mode 100% of the time (though just not leaving PA10 floating is probably sufficient).
Hypothesized Explanation: In the last paragraph of Sec 4.3 of AN2606 (the STM bootloader manual), it states:
PA10 is the RX pin for USART_1. I believe that since PA10 is floating, the bootloader senses traffic on USART_1, and believes that's the interface it should use, and thus turns off USB. By tying PA10 to GND, the bootloader doesn't sense UART traffic and correctly enables USB.It is recommended to keep the RX pins of unused bootloader interfaces (USART_RX, SPI_MOSI, CAN_RX and USB D+/D- lines if present) at a known (low or high) level at the startup of the bootloader (detection phase). Leaving these pins floating during the detection phase might lead to activating unused interfaces.
I've started a new thread for this issue, since it seems important enough to warrant its own discussion. I believe my problems may be exacerbated by being in a climate with low humidity, since it seems like 99% of people don't have this issue.