I seem to have it sorted now.
From your link, I made the Group Policy change as suggested. The FRDM-K64F has the OpenSDA v2 bootloader, so the file system looks different and it didn't even show any version information, just a bootload.htm file that contained nothing but 512 nulls.
But I was able to recover without a probe.
Starting from OpenSDA Serial and Debug Adapter|NXP, I downloaded http://www.nxp.com/files/32bit/software/CMSIS-DAP-BOOTLOADERV2.zip and copied the bin to the card (which was already stuck in bootloader mode) during my 30-second window when I can see the device. This at least brought me back to my original state where the MBED device and serial port appear but reset every 30 seconds.
Then I downloaded the latest firmware application from Firmware FRDM K64F - Handbook | mbed (version 0226) and copied it to the card.
Now it works.
Meanwhile, since it might have been a hardware problem, DigiKey sent a replacement card which arrived today. The new card had the same initial problem. But I was at least able to see that the DETAILS.TXT file said it came with version 0201. The same solution of updating the bootloader and application worked again.
So it seems like it was something about how the boards were originally flashed and maybe a conflict with Windows 10, despite having made the Group Policy change before connecting the second board for the first time. The serial number on the second board was exactly one higher than the first board, so I assume they were flashed the same way.
Thanks for your help.