TL;DR: Bad soldering. A really dumb mistake that I haven't discovered it until observing an anomaly on the oscilloscope a week later.

Spent several more days on diagnosing the I²C issue with the ESP-WROOM-02 module. Finally, I managed to get access to an oscilloscope. I hooked up the SDA (CH1) and SCL (CH2) lines, and the root cause became obvious: The SDA line could not be driven low!

At this point, I was skeptic if the boot pins (i.e., IO0, IO2, and IO15) cannot be used as GPIO (and hence the bit-banging I²C), so I tried running a simple sketch to toggle IO2 (CH1) and IO15 (CH2) as shown below.

Obviously there was something wrong with IO2. I repeated the same experiments on an ESP-12F breakout board, and both the toggling and I²C (IO2 as SDA and IO14 as SCL) functioned flawlessly.

IO2's waveform tells that the firmware attempts to drive the pin low, but something resists strongly. Then I looked for a short between VCC and SDA. Bingo.

Fixed the short, and got the BME280 and VEML6075 sensors working. After all, I could not trace the origin of the short, but my bet is on some tiny DFN or LGA pads...