I may have more to post in the future, but here is a first:

Already in the ancient era (as Proclus notes in his comments on Euclid) there were a few people who thought construction in geometry as being bypassed when the premise seems to allow you to do things in a single move.

A good example of this would be the second proposition in the first book of Euclid (pic provided - I actually modeled this in Blender

), where Proclus (5th century byzantine from Constantinople) says that some people just used the compass to secure a radius of BC, then created a circle from A.

But it seems that Euclid's reason for presenting this so early on was exactly to highlight that such a thing wouldn't be a valid construction, and in logic it's known as "begging the question" (petitio principii). This becomes important, since likewise in formal systems (computers and other stuff) you can't just go outside the system and derive a new formula, even if it is obvious outside the system; you have to mention either specific theorems or alter the previous statement by one of the allowed rules. It's the same here: you have to construct BC from something else (despite using a radius BC to get there), and just using the compass to move length BC around would be analogous to examining a statement from outside the system.

(the circles from A and B, with radius AB, are how to construct the equilateral triangle; the other vertex is where those circles intersect- proposition I)

(I also provided an alternative construction, without use of Proposition I but still using radius BC only from a center in known points of BC)