I think you are misunderstanding the datasheet and the need for the pull up resistor in the ESP-01 module. Or I misunderstood entirely your question. But, lets give it a try.
The ESP-01 is a full setup module and ready to run in a normal operation. The pull ups and pull downs required to bring the module to a normal operation are already in place. If you take off your circuitry and use a programmer like this ones https://www.ebay.ca/itm/203304682591, https://www.ebay.ca/itm/203304682591, you will be able to load code and have it it running just applying power.
Without going deeper in your design, I imagine you are using the GPIO2 to drive the OPTOTRIAC. And for that, you will have code to set GPIO2 LOW or HIGH according to your needs. So, the GPIO2 will never be floating. It will be either HIGH or LOW. For that, you dont need any any pull up resistor as you already setting in the code the state of the pin.
I am under the impression you are looking to the ESP8266EX IC datasheet. Or any hardware recommendation guidance for the ESP-12E/F module. Which would be another story entirely.
By the way, what is happening is not that the module is not going to normal operation. The module is working as expected by the circuitry. But, the pin is held HIGH and you are under the impression the module is not working. If you put something in the GPIO0 you will probably see that it is still working.
In other note for the design you implemented, it is not that your circuitry is working when you go lower in the resistors. What you are doing is creating a short. The way I see from the image, you are always setting the OPTOTRIAC to HIGH. When your PIN is HIGH, the value from your PIN to the OPTOTRIAC will be high just because all nodes are HIGH. But, when your pin is LOW, the OPTOTRIAC still going to be HIGH as there is a connection from it to 3V3 through the resistor.
The fact that is working with lower resistor is because you are creating a short from 3V3 to GPIO2, when you set the PIN to LOW and the resistance is LOW. So, all the current will be sunk to the PIN instead of going to the OPTOTRIAC. And that will cause it to turn OFF giving the impression that this is working. It is just a fake impression that it is working. But, I believe that is not what you wanted to do.
So, take the resistor off and give it another try. The OPTOTRIAC will follow whatever you set in the PIN via code.
Take a look at this video as it goes all the way to the IC on ESP8266