I think it probably wouldn't have been much of an option, or at least not worth the trouble (even for AB-).
Most likely the patient died while on some sort of painkiller or anesthetic (morphine, in most cases), and I think any of that would make it less than ideal.
You might have diseases in the blood, which you would have to run tests on somehow after drawing all of it.
I also found, after a couple minutes' research, apparently death causes stasis of the blood within minutes, leading to the endothelium breaking down (it's part of the vessel wall). That may compromise the contents of the blood by the blood melding with the nearby tissue, I suspect.
And, of course, blood cells need oxygen to survive. That probably leads to them dying within minutes, or possibly. Of course, that makes me wonder why donated blood cells live on in a bag for so long--some sort of artificial coagulant, I guess.