James,
I'm not repeating old doctrine. More coherent logic lies in the specification I'm taking great lengths to repeat.
You say "I don't need to think 'very hard' about something so simple." Yet you'd failed to take note so had missed the point yet again;
This really must be the last time I try! Perhaps forget the near/far field TZ and just focus on this. The solution IS ostensibly simple but it's the implications that matter. The DFM identifies that are TWO cases of energy emissions;
1. A pulse of a certain wavenumber, say 10 is emitted taking 1 second.
2. A continuous stream of fluctuations is emitted, at 10 per second.
First we consider A) the kinetic effect. If the observer is at rest wrt the emitter he detects them both over 1 second, but if he is in motion (say towards the emitter) the two cases vary. In case 2, the continuous wave, he times it for 1 second (for 'frequency) and finds it blue shifted and, guess what; "Higher Energy"!! But that is only because he now has MORE WAVES jammed into that 1 second (or less if he's receding). There is no 'change' in energy only the relative times.
Now case 1 of the short 10 wave pulse. He again times it to find frequency, and finds the SAME frequency as case 2! Now what present text books have missed is that the energy in case 2 is actually the same as emitted, just squashed into less time, so when a second in the NEW frame is used, the energy appears higher. Case 1 just gives us a 'control' case to analyse it with.
What that means is that the Doppler shift equation we use in theory, which is shift of 'frequency' makes nonsense of everything. We need to know wavelength through the background frame, which is not available to us. We can however calculate it if we know OUR OWN speed wrt the emitter (or background propagation frame) and the new 'wavelength' propagating in our own rest frame. Current theory doesn't bother with any of that so ends up as nonsense.
The case for a gradually changing medium density is different again as speed between particles is always c, and just the 'number of delays' changes. Wavelength then also changes, as does 'apparent/global' propagation speed. Again a fool not working on lambda and wavenumber would simplistically assume it is the emitted 'energy that has been changed.
The 10 wave parcel REPRESENTS a time signal of 1 second. If it dilates it does't mean 'time' has mysteriously been altered somehow! It's only because the mechanical signal sequence has been contracted or dilated.
Understanding each bit of that is not the battle, it's applying it across all current beliefs and seeing how they must change. Maxwell equations are greatly simplified, as given in my essays. I won't do this again but if you haven't rationalised it all yet don't worry, it took me months to unravel the knotted mess, just keep re reading, plus my essays. (but don't 'read over' any). I hope the hidden simplicity emerges.
best wishes
Peter