@NigelJ and I got into a comment discussion on this question. I probably shouldn't have started the discussion, but I wanted to respond to an apparent frame challenge by challenging it. The interchange of comments, though, especially the comment "I cannot answer on behalf of [Protestants believing differently than NigelJ]", made me think. If different Protestants can (apparently) have different views on whether the question is even phrased correctly, is it appropriate to tag the question "Protestantism"?
I think the question is fine as a question of church history. I don't think anyone would dispute the basic facts: that before the reformation infant baptism was supreme in the Catholic Church, that most early reformers still baptised infants, that the anabaptists did not and would even rebaptise adults, and that since then both Reformed and Arminian baptists have grown often without a lot of direct links to the anabaptists. A good answer would trace how it spread from anabaptists to other Protestants even without being direct descendants from the anabaptists, or in other words, how the idea spread sideways between the branches of descent among Protestant denominations.
The question itself is actually not too bad, but it is a little off in that it pretends this is something that there is going to be an objective right answer too no matter what your viewpoint. This expectation itself is objectively false given thousands of years of disagreement over the "right" answer. If this question was easy to answer as framed we wouldn't be where we are now.
The issue with answering this is going to be similar to NigelJ's comments: dogmatic assertion that only one side of the story even exists. (Sorry mate, I'm not trying to put you down here, but implying nothing else is there to be seen in Scripture is pretty dogmatic for an issue that solid folks all round have seen other sides to for a couple thousand years.) In fact I'll say that even I would have a hard time answering the question as worded, because it would be almost impossible to do so without asserting my own perspective as the way it actually happened. However much I think that to be the truth, clearly it is a contested issue (case in point, I see the opposite story as NigelJ and think its relatively obvious when you sum up the whole Scripture story).
All that to say I think the question is going to be a hotbed for "truth takes", and would be much better directed at either credo- or peodo- baptists. Ask two questions if needed, but 'protestantism' is indeed too broad a scope to frame this question.