All this is obscured for us by the fact that for many generations, there were always more adults joining the church than children of Christians to be brought up in church. And there was also a time in the 3rd & 4th Centuries when many believers delayed baptism until their deathbeds (or call to ministry) because they were afraid of being punished for sins committed after baptism. Once the church had worked out a consistent means of confession and penance (public and private), this worry faded away. By the High Middle Ages, the church was running out of peoples to evangelize and churches were everywhere. At that point, the baptism of children, which had been common all along, perforce became the norm.
"Believers' baptism" is not a return to apostolic doctrine or practice. It is, in fact, an entirely new way of understanding and organizing the conversion process which arose in the 16th Century. The Anabaptists began it (and they sprinkled at first; they took over immersion as the only proper mode from the anti-Trinitarian Socinians). That doesn't mean that believers' baptism is a bad way to do church, for those who want that. But the idea that it is simply right, and those of us who baptize once only, by whatever mode, at whatever age, are wrong, is bunk.
Many years ago, I had an old lady argue baptism with me. She thought immersion was the only authentic way to baptize. I got tired of her polemic, so I made her an offer. I told her that henceforth, I would agree that in that congregation (only), I would wait to baptize only on profession of faith by immersion, IFF she would agree that we would use real wine in communion. 'Cause I can make a heckuva stronger case for using real wine in communion than I can for believers' baptism. She pursed her mouth tighter than a miser's moneybag and didn't answer me.