“It took off primarily through the efforts of Constantine, who made it the official religion of the empire”
Only in part. Christianity had spread through about 20% of the Roman Empire before Constantine’s conversion. This kind of a spread from a non-militant religion that originated in the lower classes is historically unheard of.
“Christianity got a head start of over 600 years to Islam, yet Islam now has over 1.5 billion followers.”
True. And much of Islam’s spread has been by the sword. Some would argue the divine nature of Islam’s conquests as its proof.
My simple point is that a lower-class religion spread in a way that’s really unprecedented in history. It did so without armies, without power, and relatively without influence.