Hi there,
I don’t have a spoon 
But start with ,Google and search Arduino Soft Serial and look those over, AND/OR uncomment the code you have here and change the pins to The RX and TX pins on the Expansion, D7 & D6. Also I doubt the Radar comes set at default rate of 115200 but may be wrong. Check that.
should add some print statements after initializing and it helps check you work.
You said the hardware is the same , you are only unplugging one chip and replacing it with the other.
what about the (SWD PINS) underneath the DEV board. ARE they just touching all the stuff under there?(esp vs nrf with Expansion board info/differences) Probably not good?
HTH
GL 
PJ
I think it’s the PINS touching underneath messing things up ? LMK
If you use some "if defines" you could use the same code for both MCU's and when you compile it will know by the BSP you select.
ie. IF Defined "esp32c3"
set softserial stuff
do ESP32C3 stuff
Else IF Defined "ESP32S3"
use the Hardware Serial1 ;serial 1 port
do ESP32S3 stuff
set PSRAM etc..
END IF.
the rest of the code.
end if