Just to "yes and..."
The USOPC receives a significant portion of U.S. broadcast revenue from the multi-billion dollar deal with the IOC and NBCUniversal to air the Olympics in the U.S. It also gets some corporate sponsorships with Nike, Coca-Cola, Visa, and others for marketing rights. It raises money from individuals, including one-time and recurring donors, as well as major gifts from wealthy patrons and foundations. The USOPC licenses the Team USA brand and selling Olympic-branded merchandise. And the USOPC and its affiliated foundations host gala events, auctions, and donor summits to raise funds directly. And I'm sure specific sports national governing bodies are prominent enough to do their own fundraising if their sport is prestigious enough (gymnastics).
The obvious issue for billiards is that the BCA doesn't receive any of that financial support from the USOPC. Of course pool isn't in the Olympics, just the World Games. Which doesn't preclude it from getting some USOPC support like bowling does. I'm sure if pool got in the Olympics it would go a long way to get access to USOPC funding. Maybe even participation in the Pan American Games would help its case. But I've heard the primary sticking point is that the USOPC only wants to work with a single national governing body (NGB) for a sport and the main deal breaker is that the BCA is not a unified North American governing body for pocket billiards, snooker and carom (it's just pool). Kind of like how the IOC only wants to deal with the WCBS, not the WPA. USOPC wants to deal with something that doesn't exist, not the BCA.
Your point is valid that any debates between socialism and capitalism aren't relevant. But I thought I'd contribute what appears to be the primary deal breaker for supporting our athletes under that umbrella. It's very hard to see how American pool talent develops outside the model of pool hobbyists supported by family at a very young age become motivated to "go pro one day" and find success commercially through events and sponsors. It's hard to see how we develop a structure that systematically develops talent without a major breakthrough on the commercial side of pool.