To hypothesize, I wired up the setup as imaged below:

The idea being that the HDMI splitter
1) strips the HDCP thereby eliminating any possible encryption errors
2) has indicator LEDs on the outputs which light up when a sink is connected -- they only light up when the corresponding output is actually made active on the AVR
---> This allowed me to confirm that the Sil9387 indeed negotiates with the selected input on HDMI 1-3. Comms with the main CPU must be good then.
3) provides a means where I can easily provide an input source to the HDMI switch Sil9387, while simultaneously force-feeding the same video signal into the Sil9233.
Now, 3) seems very much like a 'random idea', but there's some rationale behind it:
* the board has no traces for the SDA/SCL between Sil9387 and Sil9233 channel -- apparently the negotiation is only performed on the Sil9387 then, and perhaps the main CPU configures the Sil9233 to a compatible mode?
* I intended on lifting the output pins on the Sil9387 so they would not interfere with the secondary output from the external splitter -- I forgot to do so before initially hooking it up as depicted above. That means both the Sil9387 and the external splitter were simultaneously driving the input to the Sil9233. Surely that's not supposed to provide any reasonable kind of diagnostic...
Wrong! This results in video output.
So, my conclusions/confirmations for today are:
- communication and power to Sil9387 appear fine
- somehow the DDC info for the inputs through Sil9387 are properly exchanged with Sil9233, possibly through main CPU
- when AVR is set to HDMI1-3, Sil9233 is properly configured to display input coming from the Sil9387
- despite all this, the Sil9387 does NOT route its selected into its output to the Sil9233
Given all of this I must address the possibility that the Sil9387 is bad after all. I'm hesitant to draw this conclusion since I've replaced it multiple times by now.
Final ramblings:
- Perhaps the Sil9387 ICs cannot sustain the thermal stress I placed on it. That would explain why multiple chips might have failed.
- Removing it (initially) took a LOT of heat, much more than (say) an older BGA laptop GPU.
- Having learned this, for my donors, I managed to remove them with slightly lower temperature by first mixing the QFP legs with a lot of fresh leaded solder, but still needed about 300°C to remove it (on a BGA rework station with proper preheating).
- I can (could) verify the Sil9387 is still good by re-transplanting it onto its donor, but then I also need a new Sil9233 for that. They're in the mail (aliexpress again, different sellers). Not my preferred continuation.
The only other explanation is that the Sil9387 is somehow properly configured to negotiate its input, but either refuses or fails to be convinced to carry the signal through onto its output.