Why doesn't the PSU chassis shield the transformer in this case?
At relatively low frequencies - and the induced signal Dave observed was at 60-odd KHz which is "low" for this purpose - magnetic shielding is difficult. At a few hundred KHz and above, Faraday type shielding is effective - surround the object radiating or receiving with a gap free conductor such as copper or aluminum. The ambient magnetic field then induces a current in the conductor that in turn produces a nearly equal and opposite magnetic field and hence the ambient field is cancelled. (Faraday's law of magnetic induction.)
As the frequency drops, Faraday shielding becomes ineffective and instead a high permeability magnetic "wrap" around the object is required that more or less shunts the ambient field around the object to be shielded. The most often used material is mu-metal and it's extremely expensive and must be handled and worked with care to avoid reducing its shielding effectiveness,
The most likely place one will run across mu-metal shields is around the CRT of an analog oscilloscope - all high performance oscilloscopes encased the CRT in a mu-metal shield to prevent nearby magnetic fields from affecting the CRT's electron trajectory.
Some very high quality audio transformers have a mu-metal shield, but it adds considerably to the cost.
You can read about mu-metal at
http://www.mumetal.com/