I think you've misunderstood the current NASA and space situation
NASA is still the primary driver of space exploration
If you need any proof, you can look no further than Blue Origin suing NASA over a lunar lander contract
The U.S. Court of Federal Claims ruled against Jeff Bezos’ Blue Origin on Thursday in the company’s lawsuit versus NASA over a lucrative astronaut lunar lander contract awarded to Elon Musk’s SpaceX earlier this year.
And later
NASA in April awarded SpaceX with the sole contract for the agency’s Human Landing System program under a competitive process. Worth $2.9 billion, the SpaceX contract will see the company use its Starship rocket to deliver astronauts to the moon’s surface for NASA’s upcoming Artemis missions.
If Blue Origin was being directly funded, they'd never dream of suing NASA for a NASA contract. NASA doesn't just fund the listed "big three". They've been funding many different startups. In fact, there was an earlier squabble in the early days of SpaceX
Back in 2004, a company named Kistler Aerospace won a $227 million contract from NASA to complete the development of its K-1 rocket and allow for the delivery of supplies to the International Space Station. The agency justified the contract on the basis that no other US company had a launch vehicle near completion. At the time, Kistler claimed that about 75 percent of the K-1 rocket's design was complete. The company had invested about $600 million over the previous decade to reach this advanced development stage.
SpaceX CEO Elon Musk greeted this announcement with anger. Although SpaceX was only two years old and would not make its first Falcon 1 launch attempt for another two years, Musk believed that NASA awarded the contract to Kistler due to favoritism. (The company's CEO was George Mueller, a leader of the Apollo Program and longtime NASA insider). Musk felt that SpaceX should have been allowed to compete for the contract.
SpaceX would contest the decision and win. This was long before NASA's bet on the Falcon program paid off
NASA faced a critical decision in 2014 when it came to selecting finalists to complete development of shuttle replacements. The agency had a finite amount of money and three finalists—Boeing, Sierra Nevada Corp., and SpaceX. Boeing, with its legacy of success and status as a long-time contractor for NASA, was the favorite. With his limited budget, NASA's chief of human spaceflight, Bill Gerstenmaier, faced pressure to select just a single contractor to ensure the program's success.
Congress still likes old-school governmental contracts
Have you ever heard of NASA's Space Launch System? If not, it's understandable. It'll get its first launch... maybe this year? The SLS system is the new heavy-lift rocket that will see NASA tackle the moon and hopefully Mars. SpaceX and Virgin Galactic can only put things into low-Earth orbit, while Blue Origin only conducts suborbital flights at present. SpaceX's Falcon Heavy could do more, but it still doesn't have enough lift for what NASA needs. And none of the rocket is reusable, so you must build a new one every time you launch.
The SLS, however, is a giant boondoggle. A recent report outlined the massive sunk costs for the project.
Martin said that the operational costs alone for a single Artemis launch—for just the rocket, Orion spacecraft, and ground systems—will total $4.1 billion. This is, he said, "a price tag that strikes us as unsustainable." With this comment, Martin essentially threw down his gauntlet and said NASA cannot have a meaningful exploration program based around SLS and Orion at this cost.
The chief problem is "cost-plus contracts"
House and Senate members told NASA to use "cost-plus" contracts, which ensure that companies involved in the development and operation of these systems receive all of their costs, plus a fee. This tends to disincentivize timely work completed within a set budget. (Remarkably, NASA was told to continue using cost-plus contracts even after the development program.)
Your true "big three" are not the contractors you mentioned (who are building some rockets with their own money) but the SLS contractors who have been doing it for decades (emphasis mine)
Since the SLS is so crucial to NASA’s lunar ambitions, the inspector general did a complete assessment of the contracts for all of the major elements of the rocket. Three government contractors — Boeing, Aerojet Rocketdyne, and Northrop Grumman — are working on the rocket; Boeing is handling the majority of the vehicle, while Aerojet makes the engines and Northrop is making boosters that will give the rocket extra thrust at liftoff. All of the contractors have experienced technical problems and setbacks, resulting in $2 billion of cost increases and two years of delays, the report said. In fact, the entire SLS program is over budget and behind schedule by more than 33 percent, compared to the baseline figures NASA gave Congress for 2019. And that will probably grow to 43 percent, the report says, as more schedule delays occur.
SpaceX was supposed to have a competitor in launching crews to the International Space Station: Boeing's Starliner. But their cost-plus capsule failed it's test flight and has yet to fly since. SpaceX has put four crewed capsules into orbit, with more on the docket. Meanwhile Congress continues to mandate the cost-plus SLS
For human exploration, Congress once again added money above NASA's request for development of the Space Launch System rocket. NASA said it only needed $2.48 billion for the SLS rocket, which has undergone multiple delays and now may launch this summer or autumn for the first time. Congress, which has long supported the rocket as it provides many jobs across many states, instead provided $2.6 billion. Because reasons.
TL;DR
Congress mostly supports Boeing, Northrop Grumman and other legacy aerospace contractors. If anything, some members of Congress oppose supporting anyone else.
