What is it about blood which causes problems which are solved by removing it?
10 C or 50 F is pretty cold for blood. I would imagine it would difficult to maintain pressure that that temperature. Cooling the blood to that level may also damage cells, regardless of the fact that it's not freezing - that's me speculating. I would also venture to guess that its faster to cool the body with readily available cold saline then run the blood through a cooling machine. Also, under the conditions they are testing the technique, the patient has already lost most of their blood. Doctors already use blood cooling machines for certain types of heart surgery, but in that situation, they have time. With this technique, time is of the essence.
especially if it's (1) approved by the living person in advance
From the article:
"The trial can only go ahead because the US Food and Drug Administration considers it to be exempt from informed consent. That's because it will involve people whose injuries are likely to be fatal and there is no alternative treatment. The team had to have discussions with groups in the community and place adverts in newspapers describing the trial. People can opt out online. So far, nobody has."
I am extremely wary yet curious about the technique.
Why are you wary? The technique has already passed animal trials and these people are going to die anyway. At the beginning of your comment you mention a concern that people will come back with demon souls or something similar that you have "learned" from watching horror films. At first I thought you were being facetious - are you actually concerned about that?
You should try reading the article, it's rather enlightening.
"It's the best thing since professional golfers on 'ludes." -- Rick Obidiah