Nope, it is that simple and saying it isn't is playing silly buggers with definitions.
Read Mein Kampf, National Socialism was, in fact, Socialism, a bit "Socialism Lite" because Hitler didn't think he had the resources for full Socialism immediately, but that was the part of the point of the push for "Lebensraum".
The point of NatSoc was that Marxism and Capitalism were both Jewish inventions and the struggle between them a Jewish plot to "Take Over the World!!!"(yes, Hitler was insane, anything new?) To fight this conspiracy Hitler decided to use a non-Marxist Socialism centered around the Nation/Race rather than Class, but it was still very much Socialism, and for all people say "but PRIVATIZATION!!!", all 13 of the corporations the Nazi's "Privatized" things to were organs of the Nazi Party, all 13 of them operated at least as much to serve the politics of the Nazi's as "For Profit", none of them were actually independent of Government Control.
Many Historians have let themselves be drawn into playing silly buggers by socialists redefining things to say "Nazism wasn't Socialism", which is part of why it can be so hard to explain to a Holocaust denier why it definitely happened because without the Socialism part you can't actually explain why Hitler was so dumb as to kill off a big chunk of otherwise productive members of society.
If you have a few hours this is a good video, there's timestamps in the description but even the minimum watch time is like an hour+