My understanding of the ATtiny13 datasheet is that although you can ISP the micro via "SPI" it actually doesn't have the functionality to operate on an SPI bus with other devices (at least not in hardware). Is that right?
Like all AVRs, the ATtiny13 uses SPI for programming. However, it doesn't have an actual SPI port that can be used in applications. Software SPI could be implemented, of course, and Atmel has an app note on the technique:
SPI is the simplest synchronous serial protocol you can think of, and since it requires only a shift register one could wonder why it isn't implemented in the ATtiny. At the same time the protocol's simplicity offers a solution: bit-banging. SPI is simpler to bit-bang than UART, where you have to worry about timing, or Manchester.
So the reason SPI is used for programming is that it's simple and reliable. The reason why it's not implemented in hardware is that it can easily be emulated in software.