Efficiency rating does not determine how complex a crossover will be.
Indeed, that is true. HOWEVER, it is worth noting that some speakers use woofers with intentionally well-controled rolloffs to lower the crossover component count. Those drivers tend, in the main, to be lower in efficiency, because damping adds mass.
As far as the claim there is more detail in speakers with simpler crossovers... I'm not buying it.
Actually, there is some truth to that statement. But of course it's not at all what the marketing blurb implies. The extra "detail" one hears from such speakers is simply response anomalies resulting from the use of efficient (which often means somewhat raggedy native performance, especially at the ends of their intended bandwidth) drive-units with inadequately-designed (aka "minimalist") crossovers. So yes, there are parts of the spectrum emphasized (generally, upper mids) that correlate with detail perception. And because they sound different, a person can legitimately prefer that sound. That it's not what was actually encoded in the recording is neither here nor there if someone likes that particular emphasis.
However, it's unfair to tie all efficient speakers with that brush. The very best speakers, IMO, combine high-fidelity design/voicing (perhaps with a top-octave rolloff), constant directivity over a fairly narrow angle, high efficiency, and crossovers designed to contain no more (or fewer) parts than required to achieve the desired voicing.
Look at the GedLee line as an example of what can be done by people who know what they're doing.
Or some of the bigger Tannoy studio monitors.
Efficiency matters, because low efficiency speakers suffer from power compression at domestic living room SPL. Consider that a speaker that's 6dB less sensitive has to contend with 4x the heat to produce a given SPL. That heat changes the parameters of the drive-units.