So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a consistent and complete foundation of mathematics by showing it to be derivable from logic) as a result of them being shown to be impossible as a result of Gödel's incompleteness theorems.
However, I still do not understand why this does, to such an extent, defeat Logicism, whilst still allowing for the use of, for example, ZFC. Whilst I do have a vague understanding of this being as a result of how the Principia seems to lack the distinction between provability and truth, as apparent from this quote from a paper from the bulletin of symbolic logic:
"Whitehead and Russell fail to distinguish between the concept of truth and that of provability."
I still cannot precisely understand what it is, in this lack of distinction, that makes Logicism so vulnerable to the incompleteness theorems.