As far as I can tell the ESP32 doesn't have an EEPROM. It has only ram on-chip, with flash connected to store the program. Any EEPROM would be additional hardware connected to it, so I'm unclear on what you're referring to.
Are you certain that the problem, whatever it is, couldn't be caused by random gibberish being received by the serial port? A floating RX line will pick up noise from the environment, which would often be interpreted as random characters - this might cause misbehavior of the application depending on what it does with incoming serial data.
I would pull an idle serial line high with that 100k resistor, not low - an idle serial line is held high by the transmitter, so given a need to keep it in a defined state, I would pull it high, so it looks like an idle transmitter (unless you want the software to be able to tell if something is connected to the serial port, in which case that would be perfect, as you could read the pin and if it was low, and stayed low for the time it takes a byte to arrive at the expected baud rate, you know nothing was connected).