The SPI might be for the touch panel only.
Nope, I don't think so. See section 6.7. You can write commands and write to GRAM data through SPI. I don't think the OTM8009A controller itself handles touch panels.
Now you're probably dealing with an LCD module including this controller PLUS an additional controller for the touch panel.
And I don't know whether the OTM8009A is accessible through its SPI interface in your module. It may have been set to MPU parallel interface only in hardware.
To check that, you need to see whether they have given access to the IM0, IM1, IM2 and IM3 lines, or if they are at least configurable via some kind of jumpers on the board...