About Microcontroller having many SPI controllers
You didn't mention a specific microcontroller but in general, yes: a microcontroller with more than one SPI controller can utilize the controllers simultaneously to the extent it is able.

Remember that many microcontrollers are single-core and can only execute a single instruction at any one time, but with DMA more parallelism can be achieved. For instance, you could instruct the DMA engine to shift out a large portion of data via one SPI controller and the immediately after initiate SPI communications on another SPI controller.

So while you will likely not be able to run the SPI controllers exactly simultaneously, for your needs they will be effectively simultaneous. If you have highly time-sensitive peripherals where SPI operations on multiple SPI busses must be coordinated at exactly the same instant, the an FPGA is typically used to implement multiple SPI controllers that are capable of operating entirely in parallel. But if you're writing a couple bytes to a relay controller on one buss and reading a few bytes from a sensor on a different buss, I assume you're OK if these transactions happen a few microseconds apart.