What part(s) do you consider especially worth saving compare to the others?
The graphics division. Under competent management and with a small- capital injection the division could really compete against Nvidia. The same cannot be said about their CPU division against Intel and the other ARM licencees. CPUs are a game too big for AMD.
And regardless if it's the CPU or GPU division that is lost, it would mean they cannot make APUs which is at the core of AMD's business. So it doesn't make sense.
Not really. They could just license the CPU IP from someone else, for example, ARM, OpenPOWER, and add the real diferentiator which is the GPU IP.
Console makers buy APUs (despite that you said XBONE/PS4 would never be APU based, remember!). They chose the best combination of CPU and GPU available. And apparently AMD's GPU leadership mattered more than Intel's CPU leadership.
I think you are bringing a red herring to this discussion (Console makers buy APU!) instead of discuss the real question for AMD management: Is it enough to have the console customers in order to sustain our CPU business? The answer to this question is certainly no, even when we expand the scope of the analysis to the rest of the embedded market. That's why AMD is trying to get back into the server market, and not double down their bet on embedded/consoles.
The consoles were basically a business opportunity that appeared to AMD, a secondary application of their IP developed to other segments, but by no means a business opportunity with enough ROI to get more priority than others, like servers.
That said, given the anemic CPU power the consoles have, don't you think that A72 or whatever succeeds it won't pack enough punch (and be far cheaper to integrate on a SoC) than anything AMD can develop with its anemic R&D budget?