It's a wide discussion, you can see arguments of all sorts. You can estabilish where it came from, what were it's contemporary philosophical and aesthetic influences, what were the material conditions that led to it, and how far back you wish to place each of them. One can go so far as to say that Corporatism is an echo of pre capitalist Europe, and the guild economies of centuries past.

Again, if you have evidence to the contrary, you can contact the author of the research paper, or you can submit your findings to a peer-reviewed academic journal. Here are some further academic sources that also support the claim that the Nazi economy was heavily engaged in privatization, and some relevant quotes:

"After the 1931 banking crisis the survival of the four German great banks was safeguarded only by a huge injection of taxpayers’ money. In return, the great banks were partly nationalized and the two worst affected, the Dresdner Bank and the Danat Bank, were merged. Re-privatization was, however, started only a few years later and finalized under Nazi rule in 1937."

Source: After the Crisis: Nationalisation and re-privatization of the German great banks 1931–1937

"There occurred hardly any nationalizations of private firms during the Third Reich. In addition, there were few enterprises newly created as state-run firms."

"The foregoing discussion is clearly corroborated by an analysis of Nazi intentions. Available sources make perfectly clear that the Nazi regime did not want at all a German economy with public ownership of many or all enterprises. Therefore it generally had no intention whatsoever of nationalizing private firms or creating state firms. On the contrary the re-privatization of enterprises was furthered wherever possible."

Source: The Role of Private Property in the Nazi Economy: The Case of Industry

Those are not strictly speaking counterexamples. I didn't claim they couldn't transition back. Capitalist systems have tendencies which lead to socio-economic and political crisis. These, pushed far enough, and without a socialist revolution, tend to culminate in something like fascism in the modern era. But these are just tendencies. Nothing says that these tendencies must absolutely always, in every circumstance, proceed to completion. That's determines by the other casual factors, the other objective and subjective material conditions. In this case, it was the geopolitical and global economic context.

As a matter of time I'm going to focus on Spain and Portugal in my answer.

The original fascist states also all went back to liberal democracy at the end of WWII. They were crushed by the liberal capitalist states (and ofc by the Soviet Union) which correctly perceived them as geopolitical rivals of the first importance by the mid-late 30s. The other fascist states were also pressured into transition back into the liberal imperialst orbit of the Cold War. In both cases, they were reintegrated as they proved unable to fully complete a world fascist counter-revolution. You are right to bring these up as very interesting cases because they are examples of how you can transition back to liberal capitalist bourgeois democracies. But this does alter the fact that there were serious changes in the political structures of both Portugal and Spain and that the supposed continuity of these transitions are often overstated.

The transition of Spain between the death of Franco in 1975 and 1978 was not as smooth or non-violent as it's popularly imagined. It was a very violent period. Fascist regimes are inherently inefficient in the long-run from the POV of socio-economic and cultural development. Further, Franco had started to distance himself from a more aggressive fascism once it was clear by 1943 that the fascists would lose WWII into order to transition back, at least in appearance, into a traditionalist, Catholic, authoritarian one-party state. It was still fascistic, but to a lesser degree and I think it had also lost it's dynamism. This was also reflected in the internal balance of power of the Spanish political regime. The more radical fascists lost influence and the Military and Church gained more influence. Instead of radical fascist mass mobilization and constant radicalizing of the populace, Franco betted on a gradual, partial de-fascicization in which the emphasis would be on technocratic governance and in which the population would be more depoliticized and deradicalized through economic growth and benefits. It remained fascistic in relation to ethnic and national minorities and especially towards the revolutionary left. But in general terms, and notably those of economic governance, it returned to a more conservative and liberal position, rather than outright radical fascist, were the latter implies a far more total, complete level of intervention in all aspects of society. Spain became heavily integrated into the Western European and Atlantic economy. It became an ultra-conservative client state of NATO in everything but name (it was not allowed to become a member). The Partido Popular are the continuation of this more liberalizing-trad-conservative wing of Falangists, wereas Vox are representative of the more radical fascist elements. It is not for nothing that they have been forming coalitions recently. But even if Vox came to power by itself, it is not clear that they would find themselves in a different situation to the Fratelli d'Italia at present, were many reactionary aspects of the country would certainly intensify, notably towards immigrants, Muslims, racial minorities and LGBT folks, but it would be limited because a fascist government, while not immediately inconsistent with fascism, does tend to contravene the liberal principles, as liberals are only one group of pro-capitalists, and there are many political positions which emphasize different forms of capitalism, notably through different governance structures over the economy, firms, capitalists, etc.

Another reason the transition was possible is because there was a recognized incentive to compromise in order to avoid another civil war, the terror of which was still very present in everyone's minds, and this was made possible because, as noted above, the more radical elements of the falangists had been somewhat sidelined and Franco had also begun a process of deradicalizing his fascist government. Apartheid South Africa was similar in many respects, in terms of reasons for liberal transition, despite the context being extremely different in many ways, most obviously when it comes to the racial dynamics. Interestingly, it's difficult to imagine Israeli doing that kind of liberal transition at this point. Imo Israel's future may well be an extremely bloody one..

It's also worth pointing out that the 70s was a far more radical time than today. There was a lot more pressure from the social-democrat European left that shaped the debates and ideological struggle in 70s Spain, again emphazing a need to transition. Also, Carter moved away somewhat from Nixon's more active support of these regimes (the US supported their military by being their main weapon's providers).

The Estado Novo was as strange type of fascism. It was not as aggressive in its foreign policy as fascist Italy or Nazi Germany, which attempted to recreate mystical conceptions of their ancient empires in a way that directly conflicted with the interests of the other western imperialist powers. It was also pulled back into the orbit. Neither Franco nor Salazar were idiots when it came to how they needed to geopolitically and economically pivot in order to ensure their survival. Salazar did something very similar to Franco, as described above.

These other fascist states, which were tolerated because they did not pose as serious a threat to the imperial interests of the US, Britain and France, and because they were willing to accommodate these other powers' interests and cooperate. However they were also undermined by their own inefficiencies, economic and political, and pressure from the external climate of a dynamic post-war trans-Atlantic economy to reintegrate themselves, at least economically, with the liberal powers.

We know that the socio-economic base transforms itself in such a way as to overcome disequilibrating forces that emerge in their social relations, especially once the latter are no longer sufficient to further developing the means of production, especially in a system of like capitalism whose basic functioning is premised on the fact of continued production of profit to incentivize production. But what about the transformations of the superstructure? My point is that as the base structure develops in this manner, it not only does so in conjunction with the the superstructure, but not only transforms the superstructure in order to reach new points of temporary stability. It is not only the base structure that evolves, moving gradually, continually and something revolutionarily into different overall dominant modes of production, but also the superstructure, in particular the political regime, which develops, and not only between base-level modes of production, but also within the same overall type of mode of production. In some cases liberal capitalism becomes fascist capitalism. In some case fascist capitalism can transition to liberal capitalism. In some cases fascist capitalism turns into outright mass-slave economies.