I have an older T-Beam V0.5 here but was somehow assuming that V1.0 has an integrated SSD1306 which is of course not the case.
(The TTGO and Heltec LoRa32 boards all have an integrated OLED display.)
So the whole issue is the result of an externally added OLED I2C display that does not behave nicely.
Not directly related to T-Beam V1.0 and it’s power design.
In that case the topic title needs adjustment.