Actually this is not true! The Civil Rights Act of 1964 did indeed force private businesses to act to end segregation:

The act outlawed segregation in businesses such as theaters, restaurants, and hotels. It banned discriminatory practices in employment and ended segregation in public places such as swimming pools, libraries, and public schools.

So yes, the government can and does literally force businesses to provide equal access to their services.

Of course, this doesn't apply to websites because the Supreme Court appears to approve of some kinds of discrimination, but not others.

That was the my initial point. The Supreme Court is fine with discrimination against LGBTQ people. But you had better believe that if someone actually discriminated against white people or Christians, they would come down against it like a ton of bricks. Because this is not motivated by an ideological belief in the first amendment, but a conservative desire to roll back rights and access for minorities they dislike.