I don't really like to use the phrase "constant voltage" here. I think that the term "voltage source." is more clear.
In the industry the two are pretty well interchangeable.
Compare how the three WELL REGARDED amps react to the impedance curve of
the simulated load: JC-1 barely, MC501 some and Ref150 plenty. Amps do
not have a constant output impedance across the power band. Coupled to a
non-constant loudspeaker impedance, the combination of the two are a tone control.
Looking
at the phase diagram for the SF Olyiii, we see it leading the 500 to 2k
region and abruptly lags @ ≈3k. Given a selection of transformer amps,
some may sound best on 4Ω, others on 8Ω [2Ω/16Ω anyone?] and some not at
all depending how their impedance reacts with that of the loudspeaker IN THE SYSTEM.
Combined with program phase and level anomalies one has the makings for a Sisyphean task of finding an ideal solution.
Ironically it was MacIntosh and ElectroVoice that led the charge back in the late 1950s to the idea of driving speakers with an amplifier that acts with a constant voltage behavior, i.e. Voltage source. This was accomplished by simply having enough feedback that the output impedance was low enough that constant voltage was possible **with most speakers available at the time**.
As time went by, lower output impedances became possible and as a bit of a self-fulfilling prophecy, necessary.
The problem with this sort of approach has to do with how humans perceive sound pressure, which is done by sensing higher ordered harmonics. While feedback is well-known to suppress distortion, its also well-known to add higher ordered harmonics of its own through a process of bifurcation. This is why we are seeing SETs and other amplifiers that have no feedback and relatively high output impedances, since:
1) you can make a speaker designed for a high impedance output in the amp and get very nice linear response
2) in a nutshell as pointed out above, making an amplifier to act as a voltage source isn't foolproof- its a "Sisyphean task". So after 60 years we still don't have the plug and play flat frequency response that was the goal so long ago. A pragmatic observer might easily come to the conclusion that we never will.