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You are right in principle... but USB and USART2 are already used (USB->PC, USART2->ESP8266 on the shim). The task was (from my point of view) to flash new FW into ESP8266 "tightly soldered" on the shim. USART1 is free (or can be made free) for this "one-shot" task -> to flash new FW.... without any problems... which you describe in your solution to do the same through USB. It is not easy task because of USB OTG is packetizied communication. I guess the only reason for this solution is that not everybody has USB-UART available. Am I right?
OK, at first I was a bit lost. I thought you meant flashing through the USB port of the pico, but in fact you are using 3 serial ports: One to the ESP-01 (on the first UART on pico), one to the PC with an extra USB to serial on the second UART, and one to the pc through the pico USB to serial. I'll try this tomorrow.