That's not quite accurate. Under EMTALA, ERs are only required to stabilize patients. Once an indigent patient is stabilized and no longer at immediate risk, the hospital is free to discharge them even if they have serious medical conditions.
There's a much more informative article available here. [1] Stabilization does not have the colloquial meaning. It's defined as, "To provide such medical treatment of the condition as may be necessary to assure, within reasonable medical probability, that no material deterioration of the condition is likely to result from or occur during the transfer of the individual from a facility, or, with respect to an emergency medical condition..."
So somebody with a serious medial condition could be discharged, but only once that condition was treated and unlikely to further deteriorate. And then there's also a bunch of other rules hospitals that accept medicare have to follow for all patients, which are similar in spirit to EMTALA. And then there are going to state rules on top of all of this. Violations are severe with penalties able to be imposed on both the hospital and the doctors/staff involved - up to and including loss of license, and they are not covered by malpractice insurance. And the courts have invariably ruled on the side of patients, so I don't think there's any doctor that's going to be looking to try to short-serve the requirements of the law. Part of the reason you can find a million negative articles about it!
I'd say it's fairly accurate. It's difficult to confidently establish "stable" in a patient with possibly feigned or exaggerated symptoms based on history and exam alone. It is generally agreed that pain management is part of managing an emergency, and so if an emergency has not been confidently excluded, it is usually part of care until a workup is concluded.
If one could know from the initial evaluation that an emergency could not possibly be present, one could certainly discharge without additional care, but that can be impossible to know immediately (even with highly trained and calibrated eyeballs).
https://www.acep.org/life-as-a-physician/ethics--legal/emtal...
Some hospitals voluntarily do more than the law requires to treat such patients.