I read that the first emperor that converted to Christianity did so because a majority of the population converted to Christianity but why would people convert to Christianity in the first place? Seeing as for a time they could have been persecuted for it.
I don't have time right now to answer your main question, but I need to address some of the text in your post. Constantine I in the 4th century, who at least publicly was the first emperor to embrace Christianity, did so at a time when historians estimate that the population of Christians within the Empire was around 10%, not a majority.
Freeman in "A New History of Early Christianity" (2009) states that scholars put the number between 7 and 10%.