For smaller companies that are just starting out, it's relatively impossible to have prior intent when people don't know who you are or what you offer. There are a number of companies who started with nothing and grew their business via online ads and this seems like a giant blind spot in the article as well as the studies that you mentioned.
But is that number statistically significant? some do, how many others tried the same route and failed, how many grew without it, etc
Experimental design, analysis of variance are such. E.g., for the farmers and from the corn fields and hog pens of Iowa:
George W. Snedecor and William G. Cochran, Statistical Methods, The Iowa State University Press, Ames, Iowa.
These methods have been widely used in the social sciences -- e.g., my wife, Ph.D. in mathematical sociology from Hopkins, got quite good with that material. The field is quite serious and mature and goes well beyond just A/B testing.
For the practical challenges of the article, academic fields closer than economics include statistics and optimization.
For the Lagrange multipliers in the article, those likely would be from the Kuhn-Tucker (Karush-Kuhn-Tucker) conditions. There without some special assumptions, e.g., having to do with cases of convexity, the conditions are only necessary for optimality and not sufficient. Generally in practice, it is more difficult to get sufficient conditions.
Yes, correlation does not necessarily mean causality. Usually showing causality needs a mechanism; in practice showing causality just from data and/or statistical methods is difficult and rare. But in practice, correlation can be powerful enough to take money to the bank.
The problem with all online advertising is that there are huge selection effects, which are hard to correct for using conventional statistical methods. So you need to do experiments. And when economists do experiment, they find that advertising effects are so small that they are hard to measure.