What is the most overlooked consideration?

The equipment’s grip on the reproduction of the emotion of the composition and performance.  +1 for @ghdprentice term - musicality and his recommendation to not focus on the usual audiophile criteria.   The prerequisite is to experience the emotion of live acoustic and amplified musical performances.  Then you can evaluate the equipment’s grip on the music compared to this benchmark.  I agree it is unrealistic to expect to bring the music hall to your listening room.  However, the equipment should, at a minimum, bring memories of the emotional experience of a  live performance, and should evoke similar emotions to the reproduction of music in your home.  Yes, it is a  personally subjective criterion.   However, if the equipment is melodious to you, you will be engulfed in the emotional experience that the recorded composition and performance evokes.  

Not to hijack since it was mentioned twice I would pose this to the repliers commenting on the "musicality"of a component..How the H do you judge a components musicality if you are considering it for purchase & don’t yet own it?
ONLY the "usual"audiophile criteria can be used at this point of the process!

This is by no means the primary consideration, but as a retired person on a comfortable but fixed income, one of the things i always consider is whether a component is desirable by others and can I purchase it for a good price. Can it be flipped quickly, without losing money or at a slight profit.

SYSTEM synergy

It may be hard to define in words, but you know when you have it (or lack it.)
A newly selected audio component inserted into your rig can create an audio performance change now ranging from poor/non/optimal to good/ ​​​​​improved.

Think all the daily AGON forum posts asking for new components recommendations.The conflicting responses range widely and all too many simply reflect personal fave bias.

Just because a touted Brand X component seemingly works well in a Respondent system, is no assurance that it will succeed in OP’s system.. “ One size does NOT fit all”…

All the things previously mentioned are considerations, I also need to see quality parts, I also much prefer point to point wiring in pre's, amps, speaker crossovers, power supplies extremely important.