The "Stalinist system of governance" is relevant insofar as the importance of authoritarianism is to the discussion of repression and executions under Stalin. However, the vast majority of executions under Stalin took place in a relatively short period of time, in the years 1937 and 1938. This is the height of the Purges / Terror that historians talk about. I think it is necessary to hone in on this time period to answer your question, because Stalin had such a central role in that campaign and because the "Stalinist system" broadly speaking is too vast and varied over time if we try to take too wide a view. The purges were, first and foremost, an attempt by Stalin to crush all opposition. In this sense, the greatest threat to Stalin was not actually from without the party but from within - people who might actually be able to successfully challenge him.

In those two years alone around 700,000 people were executed. The NKVD had a big hand in carrying out the purges and in particular a figure named Nikolai Yezhov, who led the organization starting in 1936. Yezhov was the central figure of the purges and oversaw the execution of many of those 700,000, in one way or another. He had Stalin's support and encouragement. If you need any indication as to what would happen even to a figure of Yezhov's standing and importance in the purges if they didn't have Stalin's support, then look no further than the fact that Yezhov himself was purged (executed) in an unceremonious fashion - meeting the same fate to which had had consigned so many others, after his usefulness to Stalin had been exhausted.

Did Stalin "approve" of those nearly 700,000 in 37-38? Surely he wasn't directly aware and didn't sign off on the majority of them simply for logistical reasons. It wasn't possible - the volume was incredibly high. Nonetheless, the purge was carried out under his direction and had his approval generally speaking. In this sense, Stalin absolutely deserves the reputation he's gotten for being behind so many executions.

The "Stalinist system" was also part and parcel to this process, making accusations against people both welcome and deadly - they were literally looking for reasons to purge people. But I don't think you can so easily disentangle that system from the man himself, who very intentionally put it into practice. This is of course to say nothing at all of the imprisonment, exile and other kinds of general repression that took place under Stalin and which I haven't touched on here.

Still, Stalin did personally sign off on many things. Here's an example from J. Arch Getty's The Road to Terror pg. 448, in which there is both commentary and a collection of primary documents. Here is one of those primary documents transcribed:

No. 3690 24 May 1937 To the Members and candidate members of the CC VKP(b), The Central Committee has received information implicating CC member Rudzutak and CC candidate member Tukhachevsky in participation in an anti-soviet, Trotskyist-rightist conspiratorial bloc and in espionage work against the USSR on behalf of fascist Germany. In connection with this, the Politburo of the CC VKP(b) puts to a vote to members and candidate members Rudzutak and Tukhachevsky from the party and transfer their cases to the NKVD.

Secretary of the CC VKP(b) I. Stalin

This rather unassuming document tells a typical story - fabricated charges and a turning over to the NKVD (to the aforementioned force of Yezhov) who tortured and then after a show trial, shot them both. There can be no doubt that in signing a document like this that Stalin had exactly in mind what was going to happen to them.

There is a ton of scholarship on the terror/purges out there.

An approachable one is:

Barry McLoughlin and Kevin McDermott. *Stalin's Terror: High Politics and Mass Repression in the Soviet Union. Palgrave Macmillan, 2003.

The primary source above and lots of others available in:

J. Arch Getty and Oleg V. Naumov. The Road to Terror: Stalin and the Self-destruction of the Bolsheviks, 1932-1939. Yale University Press, 1999.

Igal Halfin has also done some really interesting work on communist "autobiographies" during the purges. I don't love all of Halfin's work, but I think this one is quite interesting if you are interested in a more personal view into it:

Igal Halfin. Terror in My Soul: Communist Autobiographies on Trial. Harvard University Press, 2003.

Follow-up: How do we know that number so incredibly precisely?

I made this comment in relation to another post that asked about whether Stalin ever commuted any sentences of death or showed any mercy.

Apparently the archives demonstrate that Stalin looked very closely at thousands and thousands of individual cases. He and his immediate Inner Circle signed off – – literally in Stalin's case with his famous blue pencil -- on many many death sentences. His micromanagement of the killing machine was incredible.

Background: Dmitri Volkogonov was a Soviet General who had access to many of the secret archives. After the Fall of the SU he became a liberal, pro-Democratic supporter of the new regime. He wrote some great biographies of the Soviet Leaders. In his STALIN tome he describes the thousands of appeals that Stalin received from people sentenced to death or the Gulag.

He notes that Stalin via his famous blue pencil rarely showed any mercy or ever commuted a sentence. Nothing personal moved him. In fact he seemed to delight in tricking leading high ranking prisoners to think that if they confessed he would show mercy -- but then, of course, he did not.

Once in a while he would commute a sentence but only when the appealer made some sort of practical argument. To take one example, a factory manager was arrested but then wrote a letter to Stalin claiming that the other people in the factory who were running it now were idiots and he needed to return so it could be run efficiently. Stalin commuted his sentence and sent him back to run the factory.

https://www.amazon.com/Stalin-Triumph-Volkogonov-published-Hardcover/dp/B008TA1NYK

I don't mean to imply that he personally approved of every single imprisonment and murder...but he kept a close eye and read many case files and appeals and made many personal decisions. The evidence is strong that if you were a VIP of any kind that he made the decision to kill you or imprison you personally.