Nope. The active device, we'll use tube for simplicity, should be relatively low output, but because the headphones are so high in impedance, even an output of 5-10k is adequate - in fact the Stax headphones all have a 5.1k resistor in line between the output device and the headphone to protect the latter. Being high in impedance, the headphone needs large voltages but relatively low currents. Hence, a 6SN7GTA with its DC voltage limit of 450volt and running 7-8 mA makes a good output device - IF all its output current goes to driving the headphones. But, if you waste a good proportion of that current in the amp load, it isn't nearly as good.
Remember that the output tube drives the headphone in parallel with its load, whether that be a resistor, a choke, or a constant current load. This means that the music signal current is distributed between the headphone and the load depending on their relative impedances.
On the other hand, the output impedance is the impedance of the tube and the load in parallel. With a tube plate resistance of, say, 8k, even a perfect constant current load with an infinite impedance results in a combined output impedance of 8k. With a resistor load of, say 50k, the output impedance is about 6.9k, a negligible difference. With a choke load, the output impedance varies with frequency, being higher as the frequency rises, but mostly dominated by the plate resistance. So the output impedance is pretty similar regardless.
In order for the tube to have a reasonable standing current, the resistor has to be around 50 kilohms unless you have a very high voltage power supply. Since the impedance of the headphone is much higher than 50 kilohms, this means that most of the music signal current is wasted in the resistor. With a choke of 100H, the impedance is only 12.5 kilohms at 20 Hz, and only gets to 50k and above at 80 Hz and above, and doesn't exceed the impedance of the headphones until you get above 1 kHz. With music signals, about 1/2 the power is in the range below 300-400 Hz. Again, this means that the majority of the music signal current is used to drive the choke, and the leftovers go to the headphone. This is not only inefficient, it causes more distortion because the tube has to supply much more current. With a really good current load of very high impedance, there is practically NO current going to drive the current source, so nearly all the current goes to drive the headphones. Do a search for the thread "Output Stage Current Requirements for Electrostatic Headphone Amps" for my full technical discussion.
The bottom line is, the output load is the plate resistor/choke/constant current load in parallel with the headphone. A perfect constant current source needs no current drive so it is invisible to the output tube, which sees only the headphone load. This is the best possible situation for the output tube.
On the other hand, the output impedance is the impedance of the plate resistor/choke/constant current load in parallel with the plate resistance of the tube. Since the tube plate resistance is much lower than that of the plate resistor, etc., the output resistance is primarily the plate resistance. So a constant current load has no significant disadvantage in terms of output impedance compared to the other two options.
Finally, a constant current load does require a heat sink, so it is more expensive than a resistor, but still significantly less expensive than a good plate choke. So, the constant current load is by far the most efficient in directing the signal current to the headphone, resulting in the least distortion and most output, has no significant disadvantages, and reasonable expense.