Uhhh...
O.K one of us is confused.
Here is the way I understand it.
The equation Wikipedia gives is for
instantaneous launch azimuth. If you use that plus take into account the Earth's rotation, then with a decent assumption about final orbital velocity you should get fairly close to the correct inclination.
Except for launches due east or west from the equator, as you head downrange, your latitude is constantly changing. So if you are holding your original launch heading, then you will be at the wrong heading for the latitude you are at at any given instant. This is an error that will keep adding up as your head downrange and you latitude differs more and more from the launch latitude. You burn more and more out of your launch plane. I don't understand how this error could be smaller than launching at the proper azimuth and holding prograde, unless there is a problem accounting for the earth's rotation.
Take the extreme example. A launch due north from Alaska for a 90 degree inclination. During launch, you start heading about 0 degrees north (actually a bit west of that due to the earth's rotation) . At some point you cross the north pole. You are now heading about 180 degrees south. That's natural. But if you are holding your launch heading, would you turn back to 0 degrees north? No, because then you would be flying loops over the north pole until you run out of fuel.
Perhaps the code could keep checking your current latitude and keep updating your heading according to your equations?
I don't understand how a vessel can be in the correct orbital plane but the wrong inclination unless it is heading in exactly the opposite direction as it should be.
As a side note,
Delta IV mediums from Cape Canaveral are more likely than not launching something to geostationary transfer orbit instead of just LEO. In those cases, since the inclination has to be reduced, they sometimes
do launch eastward and the second stage burns out of plane. If you watch launch coverage with the STK animations, you can see the inclination drops when the second stage lights and goes below 28.5 degrees by seco. In that sense, the autopilot is pretty much correct