ESP8266 has built-in bootloader that supports serial programming. Unlike AVR or other microcontrollers, this bootloader is there already to begin with. We use a USB serial (CH340-based) programmer to program it for the first time. With the firmware uploaded, it can then support OTA firmware update. But USB flashing always works, even if the firmware fails to run.
The information you found about GPIO2 may be referring to the bare ESP8266 chip. We don’t ever use the bare ESP8266 chips, instead we use the ESP-12 module, which is very common and it has built-in pull up for GPIO2. Typical circuits for ESP8266 module only require pull-ups for RST, EN, GPIO0, and a pull-down for GPIO15. I’ve never seen that pull-up is required for GPIO2.
You said without that pullup uploading rarely works. I think one possibility is that maybe you are using a serial baud rate that’s too high: 230400 baud rate should be pretty reliable in general. Higher than that generally requires the auto-reset circuit: for example, with the USB programmer that I have, I can use 921600 baud rate, because it has built-in auto-reset circuit.