The USB COM port has to be released by the uploading ESPtool and can then be opened by the SerialMonitor again. That's how I understand @kenb4's response #5.
So, for quick tests where everything happens in setup() and is intended to run just once after an upload, I simply add the delay(400);
BTW: this delay can run before Serial.begin as well.
For real sketches where a quick start after reset is interesting, I know to see the full story only after using the RESET button.
Thanks for the discussion to all.