What is the most overlooked consideration?

To address the question presented, I think anyone auditioning equipment is looking for musicality, although that likely means something different to us all. I can’t imagine that equipment would sound musical to me if it didn’t image well, offer a realistic soundstage, if it wasn't  fast, if it lacked energy and detail, etc. The trap to avoid is that a lot of sins can be hidden by a sound that is veiled. Some equipment can produce a sound that is pleasant enough, but it lacks the energy and realism necessary to be a better experience over time.  
 

As was mentioned earlier, assessing the company that makes the products as a whole might be important and might be overlooked. The big “corporate” names can certainly lose their edge and, after changing hands often enough will become at best an echo of the founders’ intent- doesn’t mean they can’t make really good stuff but it’s going to be different. Every buyer of a business is looking to take cost out of the operation. Is that coming from redundant marketing and back room operations or from R&D and quality of parts? Hard to know but don’t overlook that the manufacturer may be the same in name only. 
 

likewise, some of the best stuff is made by really small operations driven by the passion and genius of the founders. Sounds good, but can they service what they sell and what happens when the founders move on? I think these are important issues that can be overlooked in the buying process. 
 

None, absolutely none, of this matters if your room is not properly treated. There is simply no way you can properly hear what even an average system is capable of without proper room acoustics, let alone the finer points of equipment sound and integration. The modest amount of money invested for a high WAF acoustical treatment will repay itself every time you sit down to listen.

We have high ceilings in out living room 12-15 ft and fully half of the total wall area is glass, one side is open to the kitchen and dining, and one wall is rough stone - the fireplace - which acts as a diffuser. Putting white 2X4 ft and 2X2 ft 2" panels on the ceiling between the exposed wood beams spaced about 2 feet between apart has been very effective in reducing the both the reverberant field and the ambient noise level in the room. It will never be a great listening room, but it is now at least functional, and the panels overhead are not noticed by the eye. In our theater (a 5.2.2 Dolby Atmos installation with a 110" screen) we did a similar ceiling treatment, and added sidewall treatments, bass traps and a rear wall skyline diffuser which utterly transformed the space acoustically. One of the sidewall treatments uses  5 1X4 ft 2" white panels stacked with 3 1X2" black panels to look like a section of a keyboard, notes C to G. Guests always comment on that as a piece of wall art, not realizing its actual function as an acoustical treatment. 

A little thought and creativity with the acoustical treatments and the colors of the covering will make the installation visually either unnoticeable, or very attractive and will do far more for the sound quality of your system than any equipment upgrade.

Wise advice !

A little thought and creativity with the acoustical treatments and the colors of the covering will make the installation visually either unnoticeable, or very attractive and will do far more for the sound quality of your system than any equipment upgrade.

One of the most overlooked considerations, from my chair, is the amp to driver interfacing, starting with getting rid of the passive crossover between the amp and speaker/its drivers. That naturally necessitates doing the crossover duties prior to amplification on signal level (i.e.: active configuration) and having a dedicated amp channel for each driver section, which in turn means amp load independency between each of these sections. This way (i.e.: both getting rid of the passive crossover on the output side of the amp and having each amp channel drive a limited frequency span) the individual amp channels will see a significantly easier load presented to them, thereby making more effective use of their power envelope and quality potential; finally control the drivers better with direct driver connection and harnessing more of their potential as well.

Next is higher speaker efficiency, or ideally no less than ~95dB sensitivity across the board to lessen thermally induced compression issues (not least as a dynamic phenomena with dulled transient response), and to aid overall ease and fluidity of reproduction. (Then there's the importance of matching directivity patterns, especially at crossovers between different driver sections for good, smooth power response).  

When combining above two (or three) aspects in speaker design, a significant bottleneck in audio reproduction has been addressed. Active configuration isn't just about removing the passive crossover as the perhaps most "visible" measure, but as well - by the same token - to make way for amp load independency in each of their sections, which in turn affects driver performance and accuracy. 

I’m always amused when audiophiles push their version of ‘musicality’ as the ultimate truth.