Why doesn't the United States fund directly the The United States Olympic & Paralympic Committee ...
We don't mix sports and politics in the US, at least not at the national level. It's a very longstanding tradition.
State and local governments are free to disagree. Some public high school and public college sports are heavily subsidized. Since Olympiads are mostly young adults (and in some cases, children), this publicly-provided training could be called a subsidy.
... and how does it fund it indirectly?
In many ways.
As noted in the question, the the US Olympic & Paralympic Committee (USOPC) is a 501(c)(3) not-for-profit corporation. That means they don't pay federal, state, or local taxes, nor do the USPOC's subsidiary organizations that handle individual sporting events. That alone is a huge indirect subsidy.
The 1998 revision of the Ted Stevens Olympic and Amateur Sports Act granted the US Olympics Committee (now the USOPC) a complete monopoly on all things related to the Olympics in the US. Having a government-recognized monopoly has lots of benefits. These include
- Deciding which subsidiary organizations handle individual sporting events (the USOC sometimes funds these organizations, sometimes taxes them),
- Deciding which media organization gets the exclusive broadcasting rights to US activities in the Olympics. This always comes with significant payments by the media organization.
- Deciding how much to charge for event admissions, swag vendors, etc. when the Olympics are held in the US. The USOPC has a government-granted monopoly; it can charge whatever they like (but charging too much would be counterproductive).
The USOPC does not build the facilities where Olympics events are held when the Olympics are held in the US. That is the job of the cities and states in which the events are held -- with a lot of help from the federal government. According to a 2000 GAO study, "at least 24 federal agencies reported providing or planning to provide a combined total of almost $2 billion, in 1999 dollars, for Olympic-related projects and activities for the 1984 and 1996 Summer Olympic Games and the 2002 Winter Olympic Games."
This is far from a complete list.