Is this article for real ??
I worked for AA Tower/Ramp Control at DFW 25 years ago. The easiest/fastest pushback is straight back. That is, if there's nothing behind the departure gate. If there's something behind the gate (gates from another concourse, storage area, whatever) then you have to angle the pushback somewhat. The less an angle you have to use, the better, but some gates it's a 90+ degree turn, like it or not. The concern on the push angle isn't making sure the tail/wingtips don't hit anything, it's the thrust blowing the area behind once the engines have been started.

This isn't some great "employee idea", it's common sense.
Perhaps the employee idea was to reduce the angle at gates where a large angle was previously used. But IMHO all it takes is one plane a year blasting the gate area behind, blowing a bag cart into another plane, and you've lost the cost advantage of all your 30 second savings.

As for your trying to visualize this, yes, an angled pushback starts out straight but turns once clear of all obstructions. The "angle" refers more about the final position of the plane at the point the tug is disconnected. Is the plane parallel to the lead-in line to the gate versus at an angle (most common at a "square" terminal building being parallel to the concourse, but perpendicular to the lead-in line).