Hypothetically, if someone has a chronic mental illness and their employer is already aware of it (and made accommodations for it) for the time they have been employed, does the employer have the right/ability to place the employee on medical leave of absence and demand that they receive counseling/therapy before allowing them to return to work, even if the employee provides a clean bill of health from a psychiatrist?
Normally, an employer can decide whether someone is allowed to go on leave or not.
If someone has a covered disability, that must be accommodated, except to the extent that the disability makes the person unable to perform a bona fide job qualification that cannot be accommodated by any practical means.
Employers have some latitude and discretion in evaluating this question and a doctor's opinion, while it is relevant evidence that an employer should consider, isn't binding on the employer. This is because physicians can differ in their opinions on a particular case, and because physicians don't necessarily know exactly what is and isn't necessary to perform a job satisfactorily. Business owners are entitled to decide what duties go with a job, and within reason, what kind of performance of those duties meets their standards.
As the case summary linked to in the comments explains: "Counseling is a medical appointment and the determination as to whether it can be required for employment is dependent on whether it is “job related” and consistent with a “business necessity” as described in the ADA."
Under the circumstances described in the question, it is likely that this requirement would be found to have been met.
Also, an employer could place an employee on medical leave without requiring counseling or therapy as a requirement to return, so long as some other performance standard was established for the employee.
On the other hand, the employer's discretion is not unlimited. If a judge or jury finds that an employee was clearly capable of performing the job in a satisfactory manner (e.g. by noting that the disabled employee outperforms many non-disabled employees in the same position who are not put on leave), then that finder of fact could conclude that the employer had engaged in illegal discrimination based upon disability, rather than legally exercising an employer's discretion.
There are, inevitably, gray areas, and no one doctor's opinion is enough to resolve that question definitively.