The German word is Nationalsozialismus.
According to the Online Etymology Dictionary, "Nazi" mimicked the abbreviation of "Sozi" or socialist as being an abbreviation of Nationalsozialist.
This translates as National-Socialism.
Left and right wing according to today's politics
In the context of the French Revolution the original left wing was about pro-republican while the right that formed as a reaction to it was pro-monarchic.
Today's political views are based on the right wing shift of power from nobility and aristocracy to capitalism in the 1800s with the change of social class structure that took place as a result of it.
So today's Left-wing politics is generally understood as aiming to address or temper any adverse effects that may occur if the economic system gets out of whack.
See "Bismarck and the emergence of the welfare state" below to understand where this kind of socialism comes from.
In terms of ideology, the left is often associated with socialism and in having Marxist influences.
In fact, there are very few socialist economies in the world in the sense of opposing private enterprise as in workers owning the means of production. This is according to a narrative saying that workers are being exploited and is considered as the far left today.
The idea that the left means more government might come from the stereotype of Marxism-Leninism, the practical implementation of which involved central planning in the Soviet Union.
It appears that this has led certain political groups to consider that anything that involves economic interventionism is just another communism or socialism despite France using the dirigiste approach until the 1980s and various historical or current examples of state capitalism.
This view may be influenced by free market ideology which is put forward as a silver bullet for a few of society's problems and sceptics have apparently come put with the term market fundamentalism to qualify the pseudo-religious views of some of its proponents.
The idea that the right means less government comes from liberalism. It would be consistent with the liberal tradition of the United States and less so in Europe where the right had authoritarian origins as being in favour of monarchy.
Incidentally, belief in freer markets corresponds with economic liberalism which is one idea of liberalism.
I am currently researching to find out when right wing politics in Europe became influenced by liberalism like it is in our time.
The socialism in National-socialism
So, to get back to question, National-socialists did have anti-capitalists among them and more specifically the Strasserists as represented by two brothers, Gregor Strasser and Otto Strasser.
However, Benito Mussolini's fascismo had a major influence on Nazism. Hitler apparently admired the 1922 March on Rome and attempted a "March on Berlin" which resulted in the failed Beer Hall Putsch and the brown shirts were inspired by the fascist black shirts.
Gregor Strasser was critical of the influence of Italian Fascism as considering it as too conservative or capitalist.
During the Night of the Long Knives Gregor Strasser and leading members of this left wing were killed.
I don't have the references about this as of yet but apparently there was harsh repression of labour unions which would be a characteristic of fascism.
So is fascism itself left wing? It's not clear as it is said to have had a "complicated" relationship with capitalism in that private property - the basis of capitalism - was supported but attacks were directed at finance capitalism. This in a context where the world was undergoing the effects of the Great Depression.
Bismarck and the emergence of the welfare state
In passing, an interesting factoid is that claims that the nazis introduced social welfare completely ignore that the statesman and monarchist Otto von Bismarck (1871-1890) had created the paternal welfare state in co-opting the programs of the liberals and socialists he opposed.
Through further influence by Ferdinand Lassalle (1825–1864) this led to the foundation of the Social Democratic Party of Germany in 1875. This is probably the origin of the expression social democracy.
What we understand as socialism today is understood as being linked to the theories of Marxism but Karl Marx criticised this development in his Critique of the Gotha Programme as resulting in a permanent authoritarian dictatorship of the bourgeoisie.
This was the beginning of the welfare state and is considered by some as an authoritarian and non-egalitarian form of "socialism".
This could explains how liberalism in the United States differs from that in Europe in that social democracy influenced progressive ideas. The progressive label itself fell into disfavour due to the Prohibition in the United States and this would have motivated Franklin D. Roosevelt to use the liberal label for his politics instead.
The national part of National-socialism
The Nazi regime is considered as an example of ultranationalism and more specifically palingenetic ultranationalism or the myth of a "national rebirth".
This brings to mind the expression Gemeinnutz geht vor Eigennutz which could be translated as "The common good before the self good".
This today would be considered as authoritarian and a knee-jerk reaction could lead someone to conclude that it has to do with collectivist thinking in that the group is more important than the individual.
Even so, there is a possible subtlety that most people may be missing as private property continued to exist unlike in the USSR and the Nazi focus may not have been the collectivist one as in placing the group above the individual but more along the lines of placing the state above the government and the people.
If so, this included protecting the state's heritage, distinct identity and culture with Cultural Bolshevism having been designed as propaganda to reject modernist movements.
So Hitler may have built a huge government machine as you say but private property continued to exist and business was subjected to the imperatives and the population was mobilised to turn it into the overpowering war machine that swept throughout Western Europe within a year.
It is safe to say that the Nazis were extreme (extremism is a political label in itself) but the notion of Third Position opposed to both communism and capitalism is said to have as a precursor Strasserism which was already mentioned above.
So either someone can say that Nazism was both a mix of authoritarian left and right economically speaking or that is was neither.
That is probably why some self-proclaimed and paranoid anti-fascists are afraid of any political "confusionism" as in ideologies uniting the right and left wing.
Nowadays, some European political parties such as the populist National Front in France are considered as being to the right without being liberal.
Also, today's Neo-nazism is often referred to as being of the extreme right wing in having kept the nationalistic, racist and antisemitic undertones.
Most people almost certainly use political labels wrong as they simply don't have the reflex of opening a dictionary and encyclopedia.
You mention radical and in politics radicalism has the specific meaning of any principles focused on altering social structures through revolutionary means and changing value systems in fundamental ways.
Populism is a view that considers that citizens are being mistreated by an elite but it can have a derogatory meaning in having to with appealing to the interests of the common people and could thus be coupled with demagoguery which is about appealing to prejudices.
Are Nazis left or right wing?
Both or neither.
Do we (media/citizen) use words like radicalism, populism and nazi
wrongly - so we don't have to argue for our standpoints?
Yes. Not so much to avoid arguing standpoints then use them as near meaningless snarl words.
Does those words become "pointless", like Orwell said about Fascism?
The words in themselves are not pointless as they have a precise meaning but they may have become useless in common usage as some people have never experienced these ideologies.
As for George Orwell's quote, it is not clear what he was getting at unless he was expressing criticism in how democracy was used as a word.
As you probably know, he came up with the idea of Newspeak in this novel Nineteen Eighty-Four with the cult slogans "War is peace", "Freedom is slavery" and "Ignorance is strength" which were used in this fictional world to manipulate public opinion.
Fascism is indeed something undesirable as it expressed contempt through politics and broke the separation of powers. The precise definition is:
A political regime, having totalitarian aspirations, ideologically
based on a relationship between business and the centralized
government, business-and-government control of the marketplace,
repression of criticism or opposition, a leader cult and exalting the
state and/or religion above individual rights. Originally only applied
(usually capitalized) to Benito Mussolini's Italy.