Early on some hints about the extermination of the Soviet Jewry (deemed especially harmful Bolshevism carriers by the Nazis), with a threat to the rest of the Jews did appear in the official Nazi press, in an ominously celebratory tone.

https://www.hdot.org/longsys/#longSys_3-D

In a lead article of the journal "Das Reich" of 16 November 1941, Goebbels returned to the subject of Hitler's prophesy of 30 January 1939 under the title "The Jews are to Blame" ("Die Juden sind schuld"):

We are now experiencing the fulfilment of this prophesy; and Jewry is inspired by a fate which, although hard is still more than deserved. Here compassion or regret is completely amiss.

3. His formulation, that "World Jewry" is "now" succumbing to "a gradual process of extermination", made clear the fate which awaited the Jews who were being deported from the German cities for the past few weeks. Two days later, Rosenberg spoke at a press conference of the Ministry of the Eastern Territories of what was to come—the "eradication"of the Jews of Europe:

There are still about six million Jews in the East and this question can only be solved through a biological eradication of all of Jewry in Europe. The Jewish Question will only be solved for Germany, when the last Jew has left German territory; and for Europe, when there is no longer a Jew left standing on the European continent—up until the Urals [...] And for this it is necessary to push them over the Urals, or otherwise eradicate them.

But such public announcements were rare and no longer appeared later, when it came to the mass deportations to the extermination camps.

The psychology behind the phenomenon you describe is succinctly explained in a speech Himmler made in Posen on 04.10.1943:

https://phdn.org/archives/holocaust-history.org/himmler-poznan/speech-text.shtml

I also want to mention a very difficult subject before you here, completely openly.

It should be discussed amongst us, and yet, nevertheless, we will never speak about it in public.

[...]

I am talking about the "Jewish evacuation": the extermination of the Jewish people.

It is one of those things that is easily said. "The Jewish people is being exterminated," every Party member will tell you, "perfectly clear, it's part of our plans, we're eliminating the Jews, exterminating them, ha!, a small matter."

And then along they all come, all the 80 million upright Germans, and each one has his decent Jew. They say: all the others are swine, but here is a first-class Jew.

And none of them has seen it, has endured it. Most of you will know what it means when 100 bodies lie together, when there are 500, or when there are 1000. And to have seen this through, and -- with the exception of human weaknesses -- to have remained decent, has made us hard and is a page of glory never mentioned and never to be mentioned.

So even as Himmler saw the extermination of Jews as a glorious, if difficult, task, he was not stupid and understood how it would have been seen even by the average German, not to mention all the others (not excluding the occupied peoples who might start asking "are we next?", albeit the concern about the Poles and the Soviets in that sense was visibly lower).

Hence all the camouflage measures at the extermination sites - like the ones documented here http://holocaustcontroversies.blogspot.com/2018/12/whats-there-to-hide-camouflage-and.html

As the war progressed, and it became clear that some of the mass graves might soon fall into the Soviet hands and be exploited for propaganda, Aktion 1005 under Paul Blobel was instituted - Sonderkommando 1005 tried to find as many mass graves as possible, make Jews disinter them and burn the remains.

The issue became increasingly sensitive towards the end of the war, as the weakened "Reich" couldn't afford to be seen as a cruel extermination machine and needed all the help it could get, or at least as few resistance as possible, which wasn't exactly helped by the reports about the extermination camps in the Allied press. That it was a pressing issue can be seen from the episode with the attempted Red Cross visit to Auschwitz. http://holocaustcontroversies.blogspot.com/2017/05/rebutting-twitter-denial-most-popular.html#redcrosscamps

On 23.11.1944 the Secretary of State Gustav Steengracht forwarded a request for such a visit to Eberhard von Thadden, the official responsible for the "Jewish affairs" at the Foreign Office. He was told by the Swiss ambassador that a visit of a camp for Poles evacuated from Warsaw by a Red Cross representative considerably contributed to calming down the world. Steengracht further reports regarding the ambassador:

Now he repeats the request that the camps Auschwitz and Birkenau are also visited by an IRC representative in order to possibly also produce a report on these camps that would help to detoxify the atmosphere.

This was von Thadden's official response:

The camps Birkenau and Auschwitz are, according to the Reich Main Security Office, absolutely out of question for a visit. Even the Reich Germans or even members of the Reich Main Security Office would be allowed to visit these camps only in special cases and mostly only after a request to the Reichsführer [Himmler]. The Reich Main Security Office considers a request to the Reichsführer himself to be futile, but leaves it up to [you].

Indeed, they knew too well that these camps couldn't be shown, despite the desire to "detoxify the atmosphere".

In April of 1945 Himmler personally met with Norbert Masur of the World Jewish Congress. He assured him that there was no extermination at all, and whatever victims there were died from typhus etc. (hence also the crematoria). At this point Himmler was trying to appease what in his mind was a representative of the "World Jewry". Obviously, boasting about the Holocaust would have been counterproductive.

Tl;dr: at the beginning of the Holocaust (the Soviet phase) some boasts about the extermination did appear, but they stopped when it came to the extermination camps; the genocide had to be hidden from the average Germans and other Europeans (Poles not included); mass graves were massively disinterred to avoid atrocity propaganda, to which the "Reich" grew increasingly sensitive as the war progressed; the proud architect of the Holocaust, Himmler was also basically one of the first Holocaust deniers.

They thought they were doing the right thing for Germany as all history was a struggle between the Aryan race and the Jewish enemy, but they were afraid that

a) the Jewish Bolsheviks who led the Soviet Union and the Jewish capitalists who led the US would take revenge

b) the German people would object due to a misguided and sentimentali respect for the value of human life.

Regarding the latter reason, they had learned this from the T4 Aktion, the killing of "unpoductive" people with disabilities, in Germany and Austria which ran more or less openly between 1939 and 1941. It had to go underground due to protests from religious leaders and victims' families. This in spite of a sustained propaganda campaign pointing out the cost of feeding these "useless" members of society. They even went so far as to include the following as an exercise in school text books:

To keep a mentally ill person costs approximately 4 marks a day. There are 300,000 mentally ill people in care. How much do these people cost to keep in total? How many marriage loans of 1000 marks could be granted with this money?

Thereafter, they resolved to "do good by stealth", as it were. Himmler explained the reasoning behind this very well in a speech to SS officers in October 1942 in Poland:

I want to also mention a very difficult subject ... before you, with complete candor. It should be discussed amongst us, yet nevertheless, we will never speak about it in public. Just as we did not hesitate on June 30 to carry out our duty as ordered, and stand comrades who had failed against the wall and shoot them -- about which we have never spoken, and never will speak. That was, thank God, a kind of tact natural to us, a foregone conclusion of that tact, that we have never conversed about it amongst ourselves, never spoken about it, everyone ... shuddered, and everyone was clear that the next time, he would do the same thing again, if it were commanded and necessary.

I am talking about the evacuation of the Jews, the extermination of the Jewish people. It is one of those things that is easily said. “The Jewish people is being exterminated” every Party member will tell you, “perfectly clear, it's part of our plans, we're eliminating the Jews, exterminatin them, a small matter”. And then along they all come, all the 80 million upright Germans, and each one has his decent Jew. They say: all the others are swine, but here is a first-class Jew. [a few people laugh] And ... none of them has seen it, has endured it. Most of you will know what it means when 100 bodies lie together, when 500 are there or when there are 1000. And ... to have seen this through and -- with the exception of human weakness -- to have remained decent, has made us hard and is a page of glory never mentioned and never to be mentioned. Because we know how difficult things would be, if today in every city during the bomb attacks, the burdens of war and the privations, we still had Jews as secret saboteurs, agitators and instigators. We would probably be at the same stage as 16/17, if the Jews still resided in the body of the German people.

Two weeks ago I collected related answers by /u/commiespaceinvader, /u/estherke and /u/kieslowskifan:

To what extent was Hitler able to keep the true brutality of his concentration camps secret from the Allies, and from the German people?.