What is the grammatical usage the phrase "for this most private of moments."?

It amounts to the same thing as "this most private moment".

this greatest of

Arcana Cœlestia. Emanuel Swedenborg - 1840 - ... to which He conjoined Divine Truth from the Human principle; concerning this greatest of mysteries more cannot be said, only that it was the very essential Divine Good and Truth in the Lord's Divine Human [principle] whereunto truth from

The Nature of Enlightenment: The Dawn of Awakening in the West P. F. Martin - 2005 - ‎ This greatest of mysteries is also the source of all phenomena including the phenomenon we call our “self”.

The Greatest Adventure: Basic Research that Shapes Our Lives Eugene H. Kone, ‎Helene Jamieson Jordan, ‎Society for American Archaeology - 1974 - ‎ ... which is the last of all the frontiers of knowledge that man can attempt to pass and encompass . We ' ll never run out of questions on this greatest of all problems confronting man , because they multiply far faster than we can answer them !

this most beautiful of

The Most Beautiful Woman in Florence: A Story of Botticelli - Alyssa Palombo - 2017 - For what better inspiration could an artist have than this most beautiful of brides on her wedding day?” “Indeed,” Marco said again.

Here are the most often found adjectives constructed with "most" most often appearing in this construction.

Here are two examples for the form of the superlative in one single word: this rarest of, this busiest of. It is a very common construction, used with many adjectives.

addition in answer to supplementary precisions needed by user FindingNemo

I I wonder if it's that we are using the superlative adjectives here as a noun, like "the rich", "the injured', but only in superlative form.

No, this is not the case; instead, this construction has to be regarded as an idiomatic one, its meaning being defined as that of a lexical item except for the changes allowed in the idiom (adj., noun). Let's examine a few examples from the Google research "this most * moment,this most * of moments" (ngram).

It can be remarked that for two of the ajectives both forms are present in the results; they are "critical" and "inopportune". It must be understood that if both forms are not present for all adjectives the reason is essentially that people are not inspired in an equal fashion by all forms. A reading of the following examples shows that there is no difference in meaning (bolded type has been added).

1. • ... only seven days after this complicated embarrassment had arisen, and when it would have been most important to secure the goodwill and co-operation of the Emperor of the French—at this most critical moment the noble Lord thought fit to ... (1864)
   • ... were not depressed by our misfortunes: the withholding a supply would produce this dangerous consequence, that our enemies would be taught to believe, that, in this most critical moment, the King and his parliament were no longer one.(1814)
   • His disciples fail him at this most critical moment of his life, and Jesus is aware that even his Father will have to turn away as he vicariously takes on himself the sins of humankind and becomes the atoning sacrifice. (2010)
   • I close my eyes, hoping it's just a trick of the light, willing it to be nothing other than sheer panic that's caused me to lose sight of her at this most critical moment. But when I open them again, there's no Ellie to be seen, no playground in view, ... (2015)
   • If ingratitude is not one of your weaknesses, I say that it is necessary that at this most critical moment, that you should stand by this Assembly, in spite of your pet theories. You will be respecting the wishes of the country as expressed in this ... (1988)

2. • ... an already enthusiastic worldwide readership with the result that the amendments and additions I have now included will help to reinforce the relevance of its message at this most critical of moments in the evolution of human consciousness. (1996)
   • The following is an overview of the organization working on the front lines the Native American language-preservation efforts. it is not an exhaustive list but one that highlights the need for maore public involvement at this most critical of moments. It includes the initiative of corporations, universities, Nativenetworks, and nonprofit organizations. (2012)
   • ... Plaza, was among the first to go out of commission since the dry cell units, which were in jars and which generated the current for the alarms, fell off their shelves and became completely inoperative at this most critical of moments. The result ... (1978)
   • The Red Dragon was cut down, William Brandon losing his life. It was at this most critical of moments that William Stanley at last decided to take a hand. He drove in from the flank on to Richard's men. Not only was the King killed, but most of ... (1977)

By reading more extensively the results of this research, for instance, those concerning "inopportune", you should finally be convinced that there is no difference in the meaning conveyed by the form with "of" and that without.

II Can we also use comparative forms? E.g. "The more enjoyable of moments", "The sweeter of fruits" does it sound natural to you?

No, those constructions are not part of English; I have never read them.

I find for instance this example;

• ref. 1, ref. 2 ... of cinnamon, 2 gallons of syrup, and 1 gallon of water. This is much sweeter than the ast. The weaker a cordial is of spirit, and the sweeter of sugar the more oil it requires to bring it up to the desired pungency, hence the great quantity of oil ...

One must not construe this form as having a similar meaning. It means "the sweeter it becomes as more sugar is being added".

I find also this other one (ref.).

•

There is in this construction no real comparison; the comparative adjective is nominalized, and becomes the head of the noun phrase "the sweeter of nature", which is to say in other words "all of that that is contained in nature and that has a high degree of sweetness" or perhaps better put, "all of the sweeter things in Nature".

III Can the noun be uncountable? E.g. "The most hardworking of staff", "The most beloved of cuisine"? –

Yes, it can; "the greatest of ease", for instance has been much used and it is still common enough (ngram); however, there might exist in this particular usage of uncountable nouns certain questions of idiomaticity; I have no way to know. For the two nouns you choose (staff, cuisine) there is no widely use adjective (ngram).