I've successfully repaired my Onkyo TX-SR606!
I ordered 10 pieces of 565-3047-ND (CAP 100UF 10V ELECT POLY RAD) and 10 pieces of 445-2865-ND (CAP CER 1UF 50V Y5V RAD) from Digikey. Total $20.41CAD after shipping and 13% tax.
I replaced the following capacitors with 100uF, matching the colored side of the new caps to the colored side of the old caps: 8072, 8096, 8602, 8607, 8609, 8162.
I added the 1uF capacitors across the following voltage regulators (do not appear to be polarized, any direction works fine): 8651, 8008, 8005, 8654, 8162.
I did not check for a blown fuse under the PCB near the transformer, and I have not performed a factory reset (and have never performed one during my entire ownership period). Everything worked immediately after adding the capacitors.
After the repair:
* OSD displays on HDMI output. Previously did not.
* Xbox 360 (HDMI) is able to output at 1080p instantly. Previously had to wait 15-120 minutes for it to handshake.
* PC (ATI 5670, HDMI) is able to output at 1920x1080 @ 23/24/60Hz now. Previously HD resolutions were glitchy and could only do 1000i @ 60Hz or 1080i @ 23/24Hz.
All you need is a steady hand and good lighting. My soldering iron is ancient and the tip is far from sharp, clean, and precise, yet I was able to do the repair. A precision tip would definitely help if you have the option.
After soldering, wiggle each capacitor's arm to ensure good contact, and touch up soldering where needed. Everything should work after this! (unless your EEPROM chip is damaged).