I don't think presidents have all that much to do with the economy (vs. the legislature). Well... maybe the most recent example is to the contrary with certain actions that historically should have been the role of Congress.
Sort of. I've done a lot of thinking about this one, and realize that it's really not just "one person" but a very large team of individuals, along with the "politics" of everything. If the President brings on a good staff, makes solid cabinet appointments, and they themself being a singular large part of the legislative process in modern years--given that it's usually fairly difficult to get 2/3rds of Congress to agree on anything--you can see the President actually has significantly more influence than you'd think.
In addition, it's the circles that person runs in and the circles that person's people run in. Do you think judge or cabinet appointments are decided on by one person? Sure, the President is a figurehead in this position and ultimately has to say yes or no, but there's a large pool of candidates out there and it's up to the staffers, maybe friends, maybe people in the know to propose those people to the President.
So while on paper the President doesn't, or shouldn't have that much power--in actuality with our current political process it's certainly much more.
Now, that said, just like the billionaires--they can only control so much. At least in the United States, there are competing interests even among the wealthy class, and sometimes shit just sort of happens--like a meteor killing the dinosaurs....or the release of decades of e-mails, videos, documents, and communications surrounding their pedophile behavior on a secret island in the Caribbean.
I disagree, but my personal belief is the economy and the government stewardship thereof work much the same way as WH40K ork technology
If that were true then why would there be such a glaringly clear example that presidential party influences economic performance?
That's true. Though in this case the administration is taking a much more active hand in the economy than any in almost a century.
For the most part the correlation between administrations and the economy is arbitrary. But in this case I would make a case that it is causative.