hey giuliomoro, small update from me.
Fist, you were right again, there was indeed a problem in the wiring. It turned out that my 3.5mm jack had a funny kind of pinout, where the tip of the TRS jack connects to the GND, while I thought it connects to either R or L channel. Resoldering solved that issue, and now the sound comes out nice and clear, even into my amplifier.
Second point, there really seems to be an issue with one of the channels being inactive. I probed the output signals with the scope, and indeed, there was a signal on only one of the channels. Like the image of my I2S scope capture, there is no DOUT data on the high phase of the WCLK, indicating that no data is being sent to the left (or right) channel. I tried it even with the Bela image, per your suggestion, and got the sound output only at one channel.
This issue has caught an attention of one of the engineers at the TI support forum, and he's helping me by reading the i2c dump files to see what is going on in the codec's registers.