Q To what degree is Simone Weil correct in arguing that the third Reich under Hitler was merely a revival of imperial Rome?

As posed in the question title and most of the body text, this needs two little chapters to address it. The subquestion in the body is substantially different and needs another one.

If you see obvious missing parts or wrong assumptions in the question you might skip to chapter 2

1 The words and basic concepts

The Roman Empire is listed as lasting from:

27 BC – 395 AD
395 – 476/480 (Western)
395 – 1453 (Eastern)

The 'German' Empires are listed as lasting from:

Holy Roman Empire: 800/962–1806
German Empire: 1871–1918
'Third Reich': 1933–1945

The fact that the 'First Reich' of the Germans contains the word "Roman" is due to the fact that they had the theory of translatio imperii, meaning that a universal empire (holy: based on and encompassing (in theory) all of Christianity) was the continuation or re-birth of the old Roman Empire of antiquity. Blessed as 'correct' by the bishop of Rome (the Catholic pope) since Charlemagne showed up in arms on the doorsteps of the city of Rome, this entity encompassed large parts of Central Europe, including present day France, Germany and Italy. This became quickly a weak form of organisation and federalised itself into progressively more autonomous and then independently sovereign parts. When first the French and then the Italian parts broke away again the name and concept stayed but was enlarge to read "Holy Roman Empire of the German Nation".

This Holy Roman Empire was suffering of old age and when Napoleon showed up in arms at the doorsteps of Vienna, where the then Kaiser resided, it fas formally dissolved. After defeating Napoleon the Brandenburg-Prussians then competed with the Habsburg Austrians for dominance, which the Prussians won, and took as an opportunity to declare themselves a new emperor, with new clothes, but an old name for a formally and legally entirely different central power state: the second Reich on German soil.

The book that advertised to try Das Dritte Reich was quite traditional in outlook and backwards oriented, but: The temptation is to see this difficult little book as an advocate for what was to come, but this gap between the ideal vision and the historical truth is virtually unbridgeable. On the eve of publication Moeller van den Bruck inserted a preface, in which he wrote that "The Third Reich is a philosophical idea not for this but for the next world.

Therefore, (Weil's) Roman Empire and the Third Reich (Empire) were not so closely connected concepts when the term Third Reich appeared in thought.

2 Weil in Hellmans explanation and Weil's view on Hitler

Now,

As Hellman describes Weil's thought in the chapter "Hitler, the New Caesar" in John Hellman: "Simone Weil. An Introduction to Her Thought", 1982:

Like Georges Bernanos, she was working toward a total, radical renunciation of the profoundest roots of fascism. Thus she was impressed by Bernanos' analysis of the nature of Hitlerism which he described as basically a return to pagan Rome.

It was fashionable for the French (and Westerners in general) to attribute the phenomenon of Hitlerism to German national peculiarities, to a mysterious movement called Nazism, or to the evil genius of Adolf Hitler. This was far less demanding and more comfortable than any self-scrutiny to discern traces of the origins of Hitlerism in oneself. And this is precisely what Simone Weil required. Her insistence was that the Roman impulse lay at Hitlerism's root cause and that Rome not only influenced the Nazis but played and plays an extremely influential part in the history, culture, and every-day thoughts of the entire West.

Simone Weil stressed that Hitler had been profoundly influenced in his youth by a mediocre book on the Roman tyrant Sulla. For her, Hitler achieved exactly the sort of greatness which was depicted in that book, and it was the very sort for which the modern Westretained "a servile admiration." It was impossible to punish Hitler because he desired one thing alone, and he achieved it: to play a part in history. For Hitler, an "idolater of history," "everything connected with history must be good." Thus, whatever Hitler was made to suffer could not stop him from feeling himself to be a great man. Beyond that, the emergence of future Hitlers (always a possibility as long as an admiration for the values represented by Hitler persisted) would not be forestalled by Hitler's punishment either: " … it will not stop, in twenty, fifty, a hundred or two hundred year's time, some solitary little dreamer, whether German or otherwise, from seeing in Hitler a superb figure, with a superb destiny from beginning to end, and desiring with all his soul to have a similar destiny. In which case, woe betide his contemporaries.
[…]
As long as Hitler was considered merely a psychopathic personality, or the sort of leader who could only emerge among "The Hun," Hitler was being addressed from a "superior" position which was, in fact, not entirely dissimilar from that which Hitler maintained. But if, as Simone Weil insisted, Hitler was not an incongruous and alien figure but a reincarnation of a persistent Western phenomenon—from Sulla through Napoleon – then a more profound and wide-ranging remedy for Hitlerism was necessary.
[…]
Simone Weil's attachment to the teachings of Jesus must be linked with her effort to understand, and resist, Hitlerism. Since she was particularly sensitive to the "Roman" element in Nazism she paid special attention to Christ's attitude toward the basic values of the empire in which he lived. Here she was struck by the way Jesus embodied an anti-Roman,anti-imperialist outlook which ran directly against the perverted nationalism and patriotisms she loathed. He had displayed a kind of pure love of country that she could admire.

That leads us to a number of interesting views of history.

Some argue that the Roman Empire never fell but was transformed into the Catholic Church, still based in Rome and organised in dioceses as they were established under the Caesars, still with a universal appeal and demand (una sancta catholica).

Hitler was indeed another "history buff" as Weil observed, and he was also quite deeply influenced by the Catholic church (Derek Hastings: "Catholicism and the Roots of Nazism: Religious Identity and National Socialism", Oxford University Press: Oxford, New York, 2009.

These views emphasise a mental and spiritual continuity – a continuity as well as this backwards orientation on 'greater times' and delayed gratification for doing the 'right' thing in the afterlife (i.e. whether in heaven or in history books or being mentioned on StackExchanges).

So no –– The Third Reich was not a "revival of imperial Rome", in the view of Weil (or Hellman). But Hitlerism displays so many specifics and a general similarity in the general traditions – alas with an emphasis on the bad ones – of the West so that she can trace a clear lineage back to Roman times. She is not alone to observe this appropriation of historical elements. It is indeed awfully popular, if sometimes incorrect to look for patterns like these. That is an explanation for the organic growth of the cancerous ideas of conservatism and fascism within Europe out of its history, but not a teleological causality.

Hitler didn't fall from the sky nor did he ascend from hell. Hitler was no singularity being, even if he orchestrated the doing of still quite singular things. He had quite some qualities of a Caesar, not in the least the madness of a caesar, but he wasn't one and he didn't represent or attempt a revival of imperial Rome.

Subquestion on Hitlers aims

Q. Did Hitler himself make a conscious attempt to think of Germany in the Roman mold, say in his book, Mein Kampf?

No. Not at all. Except for the parallels and continuities noted above. But the interesting part here is "conscious".

In Mein Kampf Rome is blamed for allowing Jews to come to German lands for the first time. For him, Rome is once big and glorious but then degenerated Empire that fell to the German invaders. That empire itself is not really well liked by him, and real glory as well as a working political system of "great men" only started when a Germanic empire entered the scene.

A large part of that book is titled "away from Rome", detailing how German Christians tried to be more independent from the pope. His positive assessment is only for some men, military success and public architecture.

But a psychoanalytical eye might delight in reading his considerations for the future central of a movement (be it Munich, Berlin or 'Germania'):

The geopolitical significance of a central centre of a movement cannot be overestimated. Only the presence of such a place, surrounded by the enchanted magic (sic!: "magischen Zauber") of a Mecca or Rome, can give a movement the strength for the duration of a movement, which lies in the inner unity and the recognition of a point representing this unity.

And in his vision for school curricula:

In particular, one should not allow oneself to be led astray from the study of antiquity in history lessons. Roman history, understood correctly in very broad lines, is and remains the best teacher not only for today, but probably for all times. The Hellenic cultural ideal should also be preserved to us in its exemplary beauty. We must not allow ourselves to be torn apart by the differences of the individual peoples of the greater race community. […] At that time we, as a young people, stormed into a world of decaying great states, whose last giants, Rome, we ourselves helped to hunt down.

All quotes in this section from AH: Mein Kampf, ⁸⁵⁵1943, own translation.