I spent all day today struggling with external Flash, but I only had more questions and did not get any good results.
The result is that both internal Flash and external Flash, DeepPowerDown mode works fine with SPI.transfer(), but not with flashTransport.runCommand(). I tried not outputting the command 0x05, which I was concerned about, and tried to make the SPI bus waveform the same as SPI.transfer(), but DeepPowerDown mode does not work.
The clock frequency was running at 2MHz until the initialization of release, begin, etc. and 16MHz after that.
The SerialFlash library you mentioned supports the DeepPowerDown command, so I loaded the Adafruit library with this as a reference, but it is too complicated for a holiday programmer.Tomorrow I will try to run the Adafruit library with SPI instead of QSPI.
Fortunately, the Adafruit library works fine if I use SPI.transfer() only with respect to DeepPowerDown.