Posting the answer to my question. I found a radio with the same part number as mine for sale. There are plenty of them, under $100.
Installed it in my V70 and it WORKED right away, everything - radio, steering wheel controls, illumination.
No dealer needed, no codes (Klaus is correct here). This surely applies to 2005-2007, and most likely to 2001-2004 as well.

Some observations:

- It seems that Volvo changed radio part numbers often, every year for sure, but maybe even within a year. HU-650 radio has at least 5 different part numbers. The radio itself seems to be identical, but I did not want to take chances and found my exact number.
- I was doing all swapping with disconnected battery. This is needed for the climate control module. You do need to pull the whole thing and unplug both the radio and CCM to do this swap.
- I was driving without the radio for a couple of weeks, no errors of any sort, the car does not care.