Can an ATX PSU power a USB port directly?

The 5V rail will most definitely be able to supply the power that you need in your setup. For most ATX PSUs, the +5v rail has the largest current capacity of all rails on the supply. Many, if not most devices will charge as soon as there is a voltage applied to their USB port, so I'd say that aspect of your setup is also good. I wouldn't worry about a device overdraw; the current control is generally built into the device rather than the charger.

The only thing to really consider is similar to what you mentioned in your question; some devices will begin to charge at say, half of their current capacity, and only ramp up to full current once they've received some data from the charger/host device. But, this is only really an inconvenience, and most devices will charge with at least some initial current. The only device I can think of off the top of my head that won't charge at all without this enumeration is a Sony PlayStation 3 controller (but I'm sure there are more).

The final point is that this really just varies on a device to device basis. Some devices will charge with max current right off the bat, a few will be limited to initial max current, and a couple might not charge at all.

Your circuit is ok. The USB connector is connected to your ATX PSU so the power limit is determined by the PSU. You can obtain the maximum current available in the 5 V line from the datasheet of your PSU. Usually there is a sticker on the power supply which gives the maximum current for each line.

Remember that modern ATX PSU need to be turned on connecting the PWR_ON cable (usually green) to the ground.