Let's begin by the correct question here:
By 1940, the pilot Jacqueline Cochran held seventeen official national and international speed records, and she earned them at a time when aviation was still so new for many of the planes she flew to be
of dangerously experimental design.
A) and she earned them at a time when aviation was still so new for many of the planes she flew to be
B) earning them at a time that aviation was still so new for many of the planes she flew to be
C) earning these at a time where aviation was still so new that many of the planes she flew were
D) earned at a time in which aviation was still so new such that many of the planes she flew were
E) earned at a time when aviation was still so new that many of the planes she flew were
Now let's talk about your analysis.
Option A - Yes, you are correct. When we say that something was so X, we must then say "that Y". So "that Y" part is missing from this sentence. However, it is not necessary to have "that" before the clause "she flew". Even without "that", the meaning is absolutely clearly that "she flew" is about the planes that she flew. Also "she flew" is a Dependent Clause here.
Option B - This one has the same idiom error as in Choice A. Also use of "that" to modify "time" is absolutely incorrect. Only "when" can modify "time". Use of comma + verb-ing "earning" is correct in the context as it says that she held the records "by earning them...".
Option C - Correct Analysis.
Option D - You are right about the incorrect idiom use in this sentence. I would say for time, "when" is the relative pronoun to be used.
Option E - Correct analysis.
Hope this helps.
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