Sorry, the burden of proof goes backwards. When you are doing something this novel and the patient dies so soon afterwards, the null hypothesis is that it had something to do with the transplant and you have to show evidence that it was not due to the transplant.
For example, here’s what Wikipedia says this of the first receiver of a heart transplant (https://en.wikipedia.org/wiki/Louis_Washkansky):
“As a result of heart attacks in 1965, approximately only one third of his heart was still functioning. In late October, he went into a diabetic coma, but regained consciousness. Once, when he was swollen with fluid and in considerable pain, his wife Ann asked him in a whisper how he was doing. He managed a smile and whispered, "I'm on top of the world." He was also suffering from kidney and liver failure.”
And that’s a case where no good alternative treatment existed, at a time when ethics committees weren’t as stringent as they are today, if they existed at all.
Having said that, I do think you're right, the cause of death should absolutely be shared, otherwise it looks quite dodgy. If he was indeed discharged with "one of the cleanest bills of health I’ve had in a long time", then how could he die so shortly after, and why would there not be more discussion around the specifics?
It’s very possible for him to have the cleanest bill of health he’s had in 6 months or a couple of years, while still dying.
I suspect that unwinding whether his death is related to the transplant or not is non-trivial. He had a few issues that would likely kill him soon. Figuring out which one did, and whether it was caused by the transplant, may be difficult.
This is similar to AIDS. AIDS almost never kills directly. Some other disease like cancer or pneumonia does the actual killing.