Industry Nicknames
As far as "being in the pocket of Big Egg" goes, I think the real threat is Chansey.
As far as "being in the pocket of Big Egg" goes, I think the real threat is Chansey.
"Big industry" is a common nickname used to describe corporate concerns in the United States which have near-monopolistic control over significant areas of the economy, which allow them to wield a large amount of control over markets, as well as influencing political decisions. To be "in someone's pocket" means this entity can readily influence the subject's behavior, whether by bribery, blackmail, legal maneuvering, threats, lobbying, social influence, financial control, or any other means.
Of the 8 industries listed, Big Tobacco and Big Pharma are nicknames that are commonly used. Big Ag is less commonly used, but is increasingly visible, as agriculture is increasingly dominated by large and powerful corporations, rather than individual family farms. While the rest of these names are purely fictional, Randall could be imagining a possible future in which these industries become big players in the political arena. The mining industry may be referred to in this context by sector, as Big Coal or Big Oil (Randall uses the term "big hole", which sounds much sillier. Most, but not all, forms of mining involve large holes.) The U.S. automobile industry was until recent decades referred to as "Detroit," later meaning only the Big Three automobile manufacturers before falling out of common usage. "Big Foot" is likely a reference to the mythical creature Bigfoot. Those who have been on the rough end of how large organizations can push not only individuals but entire communities around in a mafia-like way may take issue with a medical board being equated with such groups.
Chansey, mentioned in the title text, is a type of female-only Pokémon who carries around an egg in her marsupial-like front pouch. For Chansey the phrase "in the pocket of Big Egg" would be rather literal, except that the egg is in her pocket, rather than the other way around. Randall does not specify why Chansey would be a "threat" or why a Pokémon would be bribing people. Perhaps because, if being in the pocket of Big Egg is bad, and Big Egg is in the pocket of Chansey, then Chansey controls Big Egg and is the one to worry about.