Interesting. I modified your sketch a bit to show the status of DTR and RTS in the serial monitor screen.
Code:
void setup() {
Serial.begin(9600);
pinMode(LED_BUILTIN, OUTPUT);
digitalWrite(LED_BUILTIN, LOW);
}
void loop() {
digitalWrite(LED_BUILTIN, (Serial.rts() || Serial.dtr()));
Serial.print(Serial.rts()); Serial.print(" "); Serial.println(Serial.dtr());
delay(500);
}
CoolTerm 1.8.0 seems to work OK for both DTR and RTS.
Clicking on the bottom-right DTR and RTS indicators shows on the serial monitor screen and the Teensy LED.

Thermite 3.4 only seems to work for DTR.
Clicking the RTS indicator does not show anything on the serial monitor screen and/or the Teensy LED.

I have the feeling the issue is related to the different Windows programs.
Regards,
Paul