The one thing that an ATX PSU has over the basic design that you're looking at is AC-DC conversion efficiency. Switch mode supplies, such as an ATX supply, can be very efficient (>90%, depends on design), whereas your more traditional PSU will most likely be well below that.
Also, unless you have a decent transformer on hand, you may find that your PSU will end up costing a whole lot more.
That said, those basic designs do work. Given the current you need to pull though, you want to be sure you've got enough filter capacitance to be able to supply the current between the full-wave peaks you'll get out of the transformer at 60 Hz (since you mentioned 120V, this most likely means you're in a country using 60 Hz).
PS: The other reason is that not everyone is good at electronics, and the last thing we want to do is have someone playing with >100V and getting themselves electrocuted.
Also, unless you have a decent transformer on hand, you may find that your PSU will end up costing a whole lot more.
That said, those basic designs do work. Given the current you need to pull though, you want to be sure you've got enough filter capacitance to be able to supply the current between the full-wave peaks you'll get out of the transformer at 60 Hz (since you mentioned 120V, this most likely means you're in a country using 60 Hz).
PS: The other reason is that not everyone is good at electronics, and the last thing we want to do is have someone playing with >100V and getting themselves electrocuted.