Euclid’s Elements was used as a Geometry textbook in essentially the same way that Virgil’s Æneid was used as a Latin textbook. Neither contained exercises anything else pedagogical, and both were used in the same way, in about the same time period.
Their heyday was in the days before schooling as we know it existed. They would be used between a student and his Latin or math tutor, a person or people who worked one-on-one with the student, often at the well-to-do student’s house.
You “knew” the material once you plowed through it. Virgil (and Ovid and Horace) were simply translated by the student as a way of practicing Latin skills. The poems were translated when the tutor said they were. Similarly, the Elements were read and demonstrated to the tutor’s satisfaction. Exercises? Why would you need exercises? That wouldn’t be the objective of the course.
Heath’s versions were mainly written for mathematicians, and were certainly not used in schools.
Our current course called Geometry doesn’t have much in common with reading the Elements. For decades, Geometry was taught in two courses, Plane and Solid. Those were the days when math schooling only went to Algebra II or so, with Freshman Math at colleges being what we call Pre-calculus. My dad, high school Class of 1949 who entered Clemson Agricultural College that same year, went through mathematics like this.
The set of postulates isn’t the same: many of the modern sets of postulates are due to contemporary writers. For example, the current formulation of Euclid’s 5th (“Given a line and a point not on that line, there is one and only one line through the point that is parallel to the given line”) is due to John Playfair, and is often referred to as Playfair’s axiom. In the full-on day’s of the 1950’s and 1960’s “New” Math, Edwin Moise’s Geometry book turned the SAS triangle congruence theorem into a postulate: proving the others (ASA, SSS, etc.) given one of them is not too bad, but having to prove an initial one is pure hell.
Moise was later joined by Floyd Downs, and their Geometry book may still be in print. Alfred Posamentier and H.S.M. Coxeter soon came out with competing books, mostly sticking to the Moise postulate model with tweaks. Those tweaks included exercises, applications, word problems, career sections, computer-aided learning, and all sorts of applied things that would never pollute the pure reader of Euclid. Those that needed practical Geometry would learn it as an apprentice to a craftsman.