Yes, the 240x240 ST7789 display works without CS. But it makes it inconvenient.
I would prefer CS pin instead of the BL pin. Both would be better.
Note that VCC must be 3.3V. There is no LDO regulator on the pcb.
BL can be left n.c. or connect to a GPIO pin.
Because there is no CS pin, the controller only works with SPI mode#3.
If you have CS pin, you can use SPI#0 or SPI#3. And if a SCK pulse is missed the controller will re-sync with CS.
The Adafruit_ST7735_ST7789 library will work with the bit-bang constructor.
If you change the hardware mode to SPI#3 in Adafruit_SPI_Macros.h it will work with hardware SPI.
David.