Victory is mine!
I finally got it to work. Took a while, I had to disconnect the I2C SDA/SDL wires and have a pull down resistor on GPIO0 for the ESP-01 to start programming. I got very tired of the Arduino IDE telling me it had a programming error. I have four ESP-01 boards and they all failed consistently. After figuring out the issues, turns out that all four are A-OK.
The bamboo box is a 5V power supply with amp meter that I have plugged into a 3.3V regular on the breadboard.
The ESP-01/OLED is pulling 76mA at that very moment.
Time to build a dedicated ESP-01 programming board and just swap it in and out.