No the PI's card interface has its own hardware block in the SoC not part of the regular GPIO, and SPI is just a subpart of it, the pins cannot (IMHO) be relocated to a GPIO pin.
Also my thought about the Boot code using SPI mode, is just an assumption based on the fact that the boot code must be as small/simple as possible, and switching from the initial SPI mode to any of the faster modes is a tricky process, code that works on all SD-cards is hard to create and extensive. Thats why I assume it (the boot code) simply defaults to SPI mode.
I've never seen an implementation of booting from an EEPROM, and yes, you need to modify the PCB, or solder wires directly to the SD-card holder.
IMHO ALL forms of Flash memory can be "flimsy", in fact the more you study how it all works, the more improbable it seems it does....
But the mayor culprit for failure to me seems to be a power shutdown while the card controller is doing "household management tasks".