The most famous historians on the topic of Stalin's USSR seem to admit that their evidence comes from rumors and such:
My collaboration with the people I have mentioned was based exclusively on personal initiative and trust. I did not make use of or have access to any closed archives, “special collections,” or any other limited-access depositories and I am not familiar with any.
[...]
In the nature of things there could not be a published source for much of the information in this book; it was passed on by the victims of repression or their friends or relatives.
For no other period [the Great Purges of the 1930s] or topic have historians been so eager to write and accept history-by-anecdote. Grand analytical generalizations have come from secondhand bits of overheard corridor gossip. Prison camp stories (“My friend met Bukharin’s wife in a camp and she said…”) have become primary sources on central political decision making. The need to generalize from isolated and unverified particulars has transformed rumors into sources and has equated repetition of stories with confirmation. Indeed, the leading expert on the Great Purges [Conquest] has written that “truth can thus only percolate in the form of hearsay” and that “basically the best, though not infallible, source is rumor.” [The Great Terror, 754]
[Footnote: Such statements would be astonishing in any other field of history. Of course, historians do not accept hearsay and rumor as evidence. Conquest goes on to say that the best way to check rumors is to compare them with one another. This procedure would be sound only if rumors were not repeated and if memoirists did not read each other’s works.]
As long as the unexplored classes of sources include archival and press material, it is neither safe nor necessary to rely on rumor and anecdote.
[...]
[Footnote: The basic works on the Great Purges are uniformly based on memoir sources. Conquest, Terror, Medvedev, History: and Solzhenitsyn, the Gulag Archipelago, all rely almost exclusively on personal accounts.]
Soviet history has no tradition of responsible source criticism. Scholars have taken few pains to evaluate bias, authenticity, or authorship. Specialists have accepted “sources” that, for understandable reasons, are anonymously attributed (“Unpublished memoir of”), and treat them as primary.
[Footnote on Page: Much of the documentation in Medvedev’s and Solzhenitsyn’s works is of this form, as is much of the samizdat material. Such documentation is methodically unacceptable in other fields of history. One would be dubious about a footnote to the “unpublished memoir of the Duc de” in a work on the French revolutionary terror.
[...]
Each of the emigre and defector sources represents a variant on the vast pool of such rumors and stories [regarding the Kirov assassination], but clearly none of them was in a position to know anything about their veracity. The authors seemed to pick the stories that fit together into particular schemes, and subsequent historians followed suit.
Indeed, in the rush to support a particular scenario, scholars have been strangely selective in their use of emigre memoirs. They have accepted and used those that supported their preconceptions and ignored those that did not. Students have embraced the rumors and flawed stories of Orlov, Barmine, and Nicolaevsky while ignoring accounts that call Kirov a “conservative,” describe underground oppositionist plots in the ’30s, and argue for the existence of a planned military coup against Stalin. The point is not that these unused memoirs are any more credible than the familiar ones, but that all memoir accounts should be subjected to intense critical attention that takes contradictions into account. All claims or hypotheses based solely on secondhand gossip or rumor should be rejected according to the elementary rules of evidence.
(A LOT more here)
So, I ask: What sources do historians use to verify Stalin's supposed killings and deadly purges, besides rumors and stories?
As others have pointed out, your problem is that you cite research from the 80s when Russian archives were very hard to impossible to get into for Western scholars.
Now that the archives are open, we can utilize sources on the purges and the Stalinist killings that were not available to us as historians in the 80s.
Jörg Baberowski, one of Germany's foremost historians of Stalinism for example uses the sources of the Russian State Archive for Social and Political History as well as the State Archive for the Russian Federation. These include the files of the Soviet Politburo as well as the Central Committee and probably most importantly, the NKVD and Gulalg secretariats (GARF -- State Archive for the Russian Federation, Fonds 9401 and 9414). Form these files it is possible tor reconstruct the purges with incredible accuracy for they include all the confessions, all the notes of Stalin on who was to be purged and even a large body of death sentences signed by Stalin directly. Taken together with the personal papers of people like Molotv and others of the close circle, historians were able to a large part substantiate what was written about in the 80s as based on rumors, e.g. surrounding the Kirov assassination.
For further use of these sources, look for example to Jörg Baberowski or Anne Applebaum who are both able to use them to paint a quite vivid picture of Stalinist terror and methods.
You're citing publications from 1989 and 1985. Unsurprisingly, scholarship has advanced quite a bit since the fall of the Iron Curtain, the opening of the various archives, and particularly the work of groups like the nonprofit and very highly regarded Memorial Foundation, which works with researchers extensively.
I recommend Antony Polansky, who focuses on Jews, particularly in Poland.
I have read The Whisperers by Orlando Figes, which I'm extremely hesitant to recommend. I found it engaging, but I read it back when the info had just started coming out about his petty bullshit with trashing Robert Service anonymously. Then Memorial turned against him... I dunno, I am way skeptical of the guy and his scholarship, but I'd probably be remiss to not mention him.
My knowledge of Russian history is mostly limited to Party, economic, intelligence, and military activity 1917-1945, and after that to the foreign intelligence activities of the KGB, so I'm of limited utility in elaborating on your specific line of inquiry, but to answer your main question, the simple answer is that the books you cited are old. We know a lot more now, and can access a lot more now. Or at least, over the last two decades. I suspect it's been getting a little more difficult to do research over the last few years.