If a component is "bright", then it’s emphasizing some details at the expense of others, or against the "musical whole". Yes, some lower end components may "fake" perception of detail by pushing up treble. Then there are some components that truly convey much more than average detail overall, and also happen to be slightly bright (or perceived as such). I would place items like the Stax 009 headphones, and some of the higher-end Van den Hul and Ortofon cartridges in this category - these items also do not promote listening fatigue; they have exceptionally smooth HF response, if a bit tilted up. The bad kind of "bright" is one which hurts your ears. Sadly, I have to place most Grado headphones in the latter category.
Any kind of emphasis we’re not used to will cause us to perceive music in a slightly different way, which can in fact cause us to "notice" details that weren’t immediately apparent before. That leads to the old review trope "I noticed notes I’d never heard before!". However, I think if you go back to the prior component and listen again, you can in fact detect these "novel" details - they just weren’t emphasized to stick out until now.