Chrome extensions - Other ways to read response bodies than chrome.devtools.network?
The feature request you linked to implies that there is no support for reading either:
Unfortunately, this request is not trivial. (...) Regarding reading the Response Body: This is challenging from a performance perspective. (...) So overall, this is just not easy to achieve...
So, no, there doesn't seem to be a way for an extension to access network response bodies, except for devtools.