The 'huge government grants' thing is doing a lot of (excuse the pun) heavy lifting, as are launch facilities and ex-gov talent.
The government is a customer on Crew Dragon, it's a government 'grant' if you consider it a grant when the government pays money to a company for stationary specifically designed for the government (say, with the respective agency's logo on it). Similarly, the launch facilities are being leased and heavily customized by private companies for their requirements. You can't just prop up a launch complex anywhere, and the best spots are where the government owned ones have been built already.
Ex-gov talent is similarly complicated because there is obviously a lot of exchange between NASA and industry, especially since the space industry is growing rapidly and has high barriers to entry constraining talent availability. For example, a lot of smaller new space companies are founded by people who have previously worked at SpaceX, but it'd be weird to say that those companies are not independent of SpaceX just because that's where their founders previously gained their industry credentials from.
The important point in terms of independence is that during Apollo, NASA dictated the design in detail to companies, the company's job was to build what NASA told them to, then NASA would take ownership of it all and be responsible for running the show. The companies couldn't choose to do whatever they wanted with the designs. Now, with independent private space exploration, NASA just presents its requirements (safety, destination, crew, payload, availability etc). Companies present their proposals, NASA decides which ones fit its requirements best and promises them fixed payments for achieving specific milestones. The risk of cost overruns, failures etc is entirely on the company. The design is developed primarily by the company, and it belongs to them, NASA is essentially just along for the ride like any passenger.