Euclid himself already treats it with gloves, it has an unusually precise formulation, and is not used in the first 28 propositions of the Elements. Why? Did he doubt it? It's not like Euclid was a formalist, in the very first proof of proposition I.1 (construction of equilateral triangle), he freely concludes from the diagram that two circles with common interior points intersect at two points. This is called circle-cirle axiom by Greenberg and is not a postulate. Why is intersection of two circles "self-evident", but (effectively) intersection of two convergent lines is not? Nor does Euclid complicate the first postulate by stipulating uniqueness of a line through two points, even though it is clear from his proofs that he assumes it. The second postulate is also weaker than what Euclid actually uses.
In contrast, with parallels Euclid is very pedantic, as if trying to attract attention to them: "If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles."
Some textbooks blame other instances on Euclid's "flaws" and "gaps", but this is apparently a modernization based on reading Pasch and Hilbert into Euclid. Euclid's method was not axiomatic, it was synthetic. In other words, when he gives a geometric construction he accepts non-deductive conclusions "from the diagram", as long as the conclusions would follow from any possible diagram consistent with the construction. Listed postulates were the ones most commonly used and/or most "subtle" perhaps? In principle there was no need to list the parallel postulate at all, like Euclid didn't list the circle-circle or the uniqueness of the line.
Even if Euclid did want to list something about parallels he could opt for I.30 instead, two lines parallel to a third are parallel to each other. Much simpler. The contrapositive to this (two intersecting lines can not both be parallel to a third) is now well known and used as "Playfair's axiom". Even assuming that Euclid didn't know that it was equivalent, Ptolemy showed it explicitly later, according to Proclus, but that was not deemed satisfactory.
Proof attempts continued for over a thousand years. What proof were geometers looking for? Surely not deducing the fifth postulate "from the other four", since not much of anything can be deduced from the other four, or all five for that matter, not even I.1. As for synthetic proofs relying on "more self-evident" claims, they were offered starting with Archimedes apparently, but never accepted. Why?