Hi there,
So Interesting problem, I have tested some of them, not this specific one though.(full disclaimr) 
Which module is this? Do you have a link?
This sounds like the RD-03E is getting powered before UART on the ESP32C6 is fully ready — or it’s expecting a quiet line before responding, by power-cycling the radar likely resets its internal serial state.
You might try resetting the radar with a GPIO line after the Serial.begin(), and add a short delay, and flush the serial buffer. Some sensors are picky about timing during boot!
HTH
GL 
PJ 