I would say that sentence A
He had started visiting a church when he was cured of injuries he had suffered in an accident.
would be correct, provided that the intent is to say that he started visiting the church before, but not long before, the cure occurred. Indeed it carries an implication that the cure was due to, or attributed to, his church attendance, although context might make it clear that this was not intended. In this I agree with the comment by kandyman.
If this is the intended meaning, a change to the simple past by deleting the "had" would be a mistake, because it would change the meaning.
The modified sentence:
He started visiting a church when he was cured of injuries he had suffered in an accident.
would mean that he started visiting the church at the time of, or more likely shortly after, the cure, and carries an implication that he only started visiting the church because of the cure, although again context might show this implication not to be intended.
This sentence (A) is somewhat complex and potentially confusing, but there is nothing incorrect about it, nor would I call it poor writing. It could be rewritten as:
He had several injuries that he had suffered in an accident. Then he started visiting a church, and soon after that he was cured of the injuries.
This avoids the past perfect in the key sentence, and makes the sequence clear. But I think it is much more pedestrian writing. The revision loses the style of the example, and also removes the implication of causation, which would be a gain if the implication was not intended, but a loss if it was intended.
Now to example (B):
People had started talking about my retirement as I was in my 30s.
This is a bit unusual, but I would say not incorrect, and context might render it clear and natural. The only reason for the use of the past perfect that I can see is if people stopped talking about the narrator's retirement later. I would add that the use of "as" is at best unclear, and "when" would be better if the intent is to indicate that the talk stated when the narrator was in his or her 30s, not because of the narrator's age.
(B1) People had started talking about my retirement when I was in my 30s. But when I developed the Mustard Club at age 42, all such talk stopped cold, and I was instantly praised as a very clever person.
This added context makes sense of "People had started talking" by showing a later event which the talk preceded. In the absence of such a later event, and particularly if the intended meaning is that the talk continues through to the present moment, the simple past would be better:
(B2) People started talking about my retirement when I was in my 30s.
The later event which use of the past perfect implies can be given in extended context, or even implied, but should be present in some form, otherwise the simple past is better.