I'd like to take your opening comment as read, so far as the reason why EMF is produced in a conductor moving in a uniform magnetic field, and address the second part of your question, how to reconcile it with Faraday's Law, which as you state requires a change of flux to produce EMF in the given circumstances.
Faraday invented, I think, the homopolar generator (see picture at this site: https://en.wikipedia.org/wiki/Homopolar_generator), which certainly produces EMF as well as measurable current. My perception that there could be an assumption in your question of what constitutes a "uniform magnetic field" caused to me to check, for the first time in about 45 years (yes, I am "old"), the specifications of Faraday's original use of the Homopolar generator as he was first formulating "his"laws.
To address aspects of your question in the hope that it leads you to the answer you want, look at the picture of the homopolar generator. To me, it appears self-evident that the magnetic field at any part of the conductive disk is "uniform" (as in "unchanging"), therefore the logic, albeit perhaps simplistic, ought to indicate that according to your iteration of Faraday's Law, that his homopolar generator should not have produced any EMF due to the instantaneous constancy of magnetic field at any point on the conducting disk - except it does!
Obviously, Faraday got it right and a century and a half of brilliant people concur with it. I am almost wanting, as a result of your question, to discuss the semantics associated with "uniform magnetic field".
Of course, the other way of addressing your question is to state that there is no such thing as a "uniform" magnetic field anyway. Any measuring instrument for EMF would draw a tiny current from the moving conductor, so right away one has a magnetic field around the conductor, and as it is moving, the resultant dynamic distortion of the magnetic field means that it is no longer uniform.
And surely, if one is not measuring the EMF across the moving conductor, and the magnetic field is truly uniform, how would one know (or care) if the wire had EMF or not?.