Regarding your summary of H&N, I think that you are perfectly reasonable to accept the article within the reservations you set up.
However, I have a question for you. You wrote; “They did a good job and provided relevant evidence for many points, yet they failed to do the same with the rest.” How much should readers be willing to overlook unverified claims in a paper? The answer to this will depend on how much background knowledge is shared and on how general the authors wish to be.
Academic writing should1 make clear what is original thought in the text and what is material derived from other sources. Readers can take or leave the original thought, and they can follow up on the derived information to verify the sources. At no stage should the reader wonder about the origin of any partof the text.
At the end of the day, I think that your tentative acceptance of H&N is probably a very good response to this particular situation.
 Any uses of ‘should’ reflect my opinion–but that opinion is liable to change!