This was part of the difficulty I had with commenting on the issue. This is basically exactly how it works at my employer (although it's a combination of state law and collective bargaining agreement terms). I accrue sick leave without limit, and may use it at my own discretion except 1) I must provide my supervisor with as much notice before the fact as possible, and 2) I must provide a doctor's note if for more than a few days. The doctor's note need only be to the effect of "Nattily Dressed Anarchists on Bicycles has a medical condition." Specific details are not required unless I choose to disclose them. In fact, on the rare occasion that I've used such a note, my health care provider just hands me two copies, one which names the condition and one which doesn't.
The actual duration of my sick leave is not limited except by how much I have accrued to begin with. Catastrophic/long-term conditions needing leave beyond that have additional options available, but documentation will obviously be required by that point.
I work for a semi-government (a public corporation established by the state legislature, but otherwise self-governing) and have a union, so more fully "private" sector experiences probably differ.
edit: soft-power consideration would also limit sick leave use. I think my employer generally recognizes the idea of a couple days off for mental health, but doing so on a weekly basis might be frowned upon. By strict interpretation of the bargaining agreement, I could get away with it, but concern for the subtle or not-so-subtle wording of this year's performance review generally prevents me, etc.