The nazis genuinely believed in some of the idea from the left.
"Genuinely believed in?" Hell no. More like "strategically employed."
No, for example they believed every worker should have an inexpensive family car (VW Beetle). That all workers should have family vacation time at at nice resort each year. Well, all workers that fit a particular racial and political demographic.
The Nazis implemented social welfare programs, but only for the "racially pure" and in a way that rewarded loyalty to Hitler.
Yes and no, if they genuinely believed an idea they would embrace it, it would not be rejected because it's popular in socialist circles toot. They were absolutely practical, but not just in terms of loyalty, but in terms of what their sometimes quackery science believed would contribute to worker health or productivity.
They preached anti-capitalism early on in their history, but soon switched to supporting industrialists and crushing unions.
Nope, they supported worker unions and industrial syndicates that were supervised by the party. As I said previously, both labor and industry were subordinate to the party. It was independence, operating outside of the party, or elements of socialism and capitalism that would compete against the party, that was crushed.
They pushed for state control of the economy and mobilized the population, but in service to war-postured nationalism (i.e., fascism), not a dismantling of class structures.
In short, the Nazis were not socialists. Stop trying to insinuate that they were.
You misrepresent what I said. What I said is that the nazis would embrace a liberal socialist idea when and if it served the party in attaining or retaining power. The same was true for capitalism. That they were very practical in that manner. That the only thing they really cared about was that the party was supreme above all. To pretend they had no overlap with socialism is the falsehood.