Would you buy a pair of speakers by just looking at the measured freq. response?

No more than I'd  purchase a piece of art based on its written description. 

While it's not the only factor, frequency response is indeed one of the main considerations in my speaker procurement. Ideally, the speakers you're interested in should have a home audition period of at least 30 days with a favorable return policy. I followed this approach for my main system in the living room. I went through two rejections before finally settling on the third set, which is my current speaker. The entire procurement process took me a literal six months to complete.

Acquiring my second set of speakers for the master bedroom proved to be a lengthier process. I was particularly interested in the Burchardt S400 mkii, which is sold by a direct sales company. Although they offered a 45-day in-home auditioning policy, there was a cost of approximately 100 euros for return shipping. Despite this, encouraged by positive reviews from trusted reviewers and user feedback, I decided to give it a try. During that time period, I invested significant time and effort, more than with the first set, in making room acoustic adjustments, SPL measurements, and repositioning to fine-tune the speaker performance. After the speaker are fully broken in and a month's daily listening, I made up my mind to keep them.  It's been almost a year now, and I haven't regretted my decision since. This is how much time and effort I took to acquire speakers.  Is it my dream end-game speaker? Well, no, but it's certainly the speaker that brings me joy in music listening within my current budget or the investment I'm willing to make at this moment.

I believe it's crucial to audition the speakers in your own space. Before making a purchase, it's essential to conduct thorough due diligence by shop auditions and studying reviews and user feedback to narrow down the final list. Frequency response is a key consideration for my screening process but I would never, and I believe no one should, make a purchase solely based on that factor without conducting a home audition because the actual response performance is always room dependent.

While the ears can tell the difference in sound between a paper cone, aluminum, ceramic, soft dome vs. hard dome, is there a measurement out there that can measure these things?

The original question in this post is an important one, vitally so for an ambiguous assumption inherently made in its wording, and taken up by the many responses that follow. This assumption concerns the phrase “to listen first”, generally interpreted to mean, anywhere.

It is my sense of things that an accurate listening for change, both better or worse, can only happen specifically, a single component change at a time, in the familiarity of one’s own system; with one’s established knowledge of its specifically selected components; of amps, DACs, servers, preamps, cabling, isolation, grounding, sources, and power supplies; within the familiarity of one’s own listening space and its unique pattern of reverberant air. Each our listening abilities is built on this foundation of familiarity.

It is also rarely discussed that never do the systems at dealer showrooms or elsewhere sound as good, or half as good, than that of our own systems. Of course it can only seem so - never mind the common fact of improper set-up, the psychological bias of familiarity immediately puts us in a position of compromise in our not being as capable of perceiving the nuance of something new and unaccustomed, and in typically taking preference for the known and comfortable.

Now, never mind one’s lack of familiarity with the entire ‘system’, a simple consideration of just one component in that entire unfamiliar system in its unfamiliar room would render any personal gauge of how it sounds, irrelevant; there being­ just too many unknown variables to account for.

It is for this reason of contextual familiarity that I only seek out those reviews, either professional or by those select members of audio forums who have established systems and listening rooms to gauge their findings by. In turn, I too base my abilities to listen on specificity, on my listening muscle memory of a system I built with the resonant intimacy of a room I know, through my ears and in my listening place.

All other kinds of general ways I ‘first’ listen, do not and cannot count, in my journey to understanding what anything one speaker, component, or room sounds like.

In friendship, kevin.

OP yes…there is a German company that makes a laser scanner of the cone or diaphragm in motion in comparison to the input signal…. You might be surprised how poorly some materials do at reproducing the input…..