I think many balk at the very idea of supertweeters because they are committed to an ideology such as "phase perfection" (new name I just made up).
Phase perfectionists want zero phase shift in the audible range if not far beyond. The want total pulse coherency, with step-type pulses reproduced perfectly in the time domain.
The arguments for this go back nearly a century, since it became clear that it was difficult to design audio equipment, especially loudspeakers, without significant amounts of frequency-dependent time delays, often thought of as "phase shift" (thought time delay is a more neutral description).
Those arguing against phase perfection generally claim it wouldn't be a bad idea, but it's too expensive when it comes up against other design considerations which are considered to be more important, such as dynamic range and bandwidth.
Blind testing has generally shown insensitivity to small to moderate amounts of frequency-dependant time delays, like those caused by most well designed loudspeakers. Many designers, such as Sigfried Linkwitz, explicity claim that the phase shifts introduced by crossovers such as the 24dB per octave Linkwitz-Riley are inaudible, and it may have larger time delays than simpler crossovers.
But phase perfectionists stick to their guns, claiming they hear the difference between speakers that use true first order acoustic crossovers (such as Vandersteen and Thiel) and comparable designs. Phase perfectionists use either single drivers, often planar (such as Quad ESL-63 and its successors) or dynamic systems with 6dB/octave or "crossoverless" design. (Crossoverless designs put a high pass network, often just a single capacitor, in series with the tweeter but let the woofer(s) roll off naturally.) Often these systems are not as phase perfect as their advocates believe they are. Most speakers which claim 6dB/octave crossover only achieve something like that electrically; acoustically their performance may be anything but 6dB/octave. For example, if a 6dB/octave crossover is combined with the 12dB/octave highpass response of most tweeters, an 18dB/octave crossover results, only being staggered somewhat by the tweeter having much lower highpass cutoff frequency.
If you count products in the marketplace, or satisfied listeners, or cumulative reviews, it is clear that the high order crossovers despite all their frequency-dependant time delay, are by far the most popular. Right now the 24dB per octave Linkwitz-Riley may well be one of the most popular crossovers in expensive loudspeakers ($1000 and up). Cheaper speakers economize with 12dB per octave.
Anyway, when I set up my supertweeters last year I did indeed examine pulse response carefully, if not exactly requiring an ideal "perfect pulse" I at least wanted a brief and distinct pulse-like waveform. You get some sort of pulse picture during the Tact RCS setup. I was able to make some adjustments to the supertweeter delay to make it blend in. Though I would not claim I am achieving anything like phase perfection.
Sufficiently above it's highpass point of 14kHz or thereabouts, the Elacs are in polarity with little phase shift. It is only around the crossover point where there is some phase funniness as they blend with the upper response of the Acoustats. I would not expect the Acoustats to have phase perfection up there either; they seem to have a peak around 13.8kHz.
You could say that both supertweeter advocates and phase perfectionists claim the need for things that haven't generally been proven to be needed. And although the two approaches could in principle be achieved, most often phase perfectionists will balk at the need for the kinds of compromises which would usually be required for supertweeter integration, and supertweeter advocates would find it nearly impossible to pursue their approach and phase perfectionism at the same time.