The reason that you are having problems with all of the statements that you have quoted is that they are not actually true.
Let’s look at them in turn a see how true they are.
I understand that once an aircraft leaves the ground it does not" experience" wind and that wind only affects speed relative to ground.
Let’s suppose that we are sitting on the runway in still air and we need 50 knots TAS to lift off. The nose of our aircraft is tied to a vertical post to prevent it from blowing away. From initial still air conditions a headwind gradually increases until it reaches 50 knots. This produces sufficient lift to raise the aircraft off the ground. We are flying, but our TAS is entirely dependent upon the headwind. If the wind suddenly drops the lift will be less than the weight and so the aircraft will descend back onto the ground. The aircraft responded to the change in wind speed, so it must have been able to experience that wind speed.
I'm struggling to understand how the wind can contribute to TAS on the ground (and shorten takeoff) and then once in the air have no effect on it?
And
I just don't understand then how a headwind can contribute to TAS on take off (air flow over the wings as stated above) and then be irrelevant to airflow over the wings when airborne.
Now let’s imagine that we have taken off and are flying into a 50 knot headwind. Our ground speed is zero, so all of the lift being produced must be the result of the headwind. If the wind suddenly increases to 60 knots our lift will suddenly exceed our weight so we will rise. And our drag will suddenly exceed our that so we will move back wards over the ground. If the wind suddenly decreases the lift and drag will decreases, causing the aircraft to descend and move forward over the ground.
These effects were produced by a change in the wind speed, so the change in wind speed must have affected the TAS. Just think what happens when you are flying in gusty conditions.
I just don't get how wind speed can add to or subtract from IAS on the ground during takeoff if the wind has no effect on IAS when airborne.
In the scenario above the sudden change in wind speed would have produced equally sudden (if brief) changes in the IAS and TAS.
So none of the “truisms” that you have quoted are correct. It would however be more correct to say that once airborne:
1. The aircraft will no long experience the ground.
2. Any steady state wind conditions will not cause any changes in IAS,
TAS, because IAS and TAS are measured relative to the mass of air.
3. Any steady state wind conditions will not cause any changes in lift or
drag, because lift and drag are determined by the relative speed of the
mass of air flowing over the aircraft.
One of the great risks in being an instructor is the fact that repeatedly stating something and having it go unchallenged by the students, reinforces the instructors confidence in the truth of what he/she has just said. But repeatedly saying something does not make it true. You questions in this thread have demonstrated how this process can cause problems for the student.