It would be ok to use the simple past tense in those places.
We can often choose more than one correct verb tense to say something.
When we tell stories about the past, we often use the simple past tense to set the main point in time, to move the story forward, and to relate series of events:
This happened. Then that occurred. Then he said something. Then she did another thing.
We can use the past perfect simple (non-continuous) in your example to give more information. The past perfect simple in this case tells us that:
A started and finished in the past, AND
B started and finished in the past, AND
A started and finished before B.
In the news story, we can use our knowledge (that people usually talk to reporters after a murder) to determine that Anna's other actions happened (and finished) before she spoke to the media.
We can also easily understand that the writer is looking back into the past from the time of writing. The main point in time that the writer visits, or sees, or communicates about is the point when Anna talked to the Russian media. Relative to that moment in the past, the writer communicates about events which happened before the time that she spoke: There were several events that Anna said happened or didn't happen at an earlier time.
The use of two verb tenses provides an additional way to express that Anna's actions are anchored to a main time period (when she spoke to the media) and a referenced period that occurred prior (her experiences near the time of the murder). In this kind of context, we use the simple past to identify such a main time period, and the past perfect tenses to talk about things that happened before this main time period.
(The perfect is technically an aspect instead of a tense, but that distinction is often considered unimportant and distractingly complex for most of us to understand.) So in the examples using the past perfect simple, we know that Anna didn't see happened before Anna told (the Russian media), and that the killer struck before Anna told the media, and that Anna saw only a car before Anna told the media.
In this case, there is no problem if we use the simple past tense only, as you did in your re-written sentence. It's easy to figure out what the order of the events were.
But in other cases, it would be more useful to use the past perfect to make clear the order of events.
Here is another example of setting a main time period in the simple past, and mentioning something that happened before that time period with a perfect tense:
It was an early June morning in 1995. Helen drove along the narrow country road that led back to her parents' home, where she had spent her childhood.
The perfect tense (aspect) had grown up lets us know that the spending of her childhood happened earlier than the main time period in this "story": The time when she was driving back home in June of 1995.
If I wanted to talk more about Helen's childhood, I could continue using past perfect tenses:
And such a lovely childhood she had spent there among the horses and the trees, and in the meadows and the hills . . . .
The last time she had driven home had been two years earlier, when her beloved father had died.
If I wanted to move the story forward to the next "scene" or "main time period", I would normally use the simple past tense:
She finally reached the house, got out of her car, and ran to the front door where her mother stood ready to give her a hug.