There have been occasions when I’ve needed to do this type of thing, from time to time.
Since your files are text files, you can use an awk command.
After creating fileC.txt, and presuming that the “>” appear only in lines you want removed, issue this command:
awk ' $0 !~ />/ ' fileC.txt > fileD.txt
The !~ (no match) operation examines each line, and if “>” is found anywhere in the whole line, skip it, otherwise write-out all remaining lines. Awk is one of the foundational tools for scripting, and it operates very quickly. [Many pardons for those in the group who despise awk! 
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