The Space Shuttle was designed to be inexpensive, but in the first launch, it was actually found to be quite a bit more expensive than it was thought. The shuttle was supposed to take only a week or two to turn around, in which case it would have been a very effective launch vehicle. Instead, the tiles ended up needing replacing after every flight, which ended up taking months, instead of a week. If you can't launch the Shuttle very often, then the cost goes up enormously. As it stands, the cost to launch the Shuttle was about $450 million per mission, far higher than even the most expensive rocket today.
Post-Challenger, there were additional restrictions on what items the Shuttle could carry into space. This became an issue for the Galileo mission, for instance, that originally intended to have another stage on board, that was removed because of potential issues. There are many material limitations on what the Shuttle would carry, due to it's requirement to protect the humans on board. Also, the cost of life of an Astronaut had to be considered any time the Shuttle was launched.
As for the zero g experiments, NASA was willing to host some of those, with the request that the science all be returned to the public. Feel free to take a look at the partners that NASA had with the shuttle experiments.
In the early days of the Shuttle, as @OrganicMarble mentioned, NASA did launch some commercial satellites. Post-Challenger, it was deemed too risky to do so, and instead focused on scientific and military payloads. Afterward, NASA did occasionally launch some satellites, but mostly only for schools as a means to inspire people to the future.
NASA might some day be able to take funds, but it isn't set up right now. And I don't think anyone would actually pay NASA to launch anything, it's costs are far higher than any rocket launch provider.