Because there was a calculus for one-place predicates, Aristotle's syllogistic, roughly equivalent to monadic predicate calculus. Aristotle does discuss "relatives" in Categories, which refer to multi-place relations, or rather to objects entering them. What will later be called oblique syllogisms involving relatives is mentioned in passing in Topics. But the modern logic of relations (polyadic predicate calculus) is significantly more complicated than syllogistic, in particular, it is undecidable. A calculus for it was not worked out until de Morgan, Peirce and Frege in 1860-70s, and it required the transfer from Aristotle's term logic to propositional logic first, which was only made available by Boole two decades earlier. Ancient Stoic logic, which was propositional, did not deal with quantification and was largely lost during middle ages, although Leibniz showed interest in it. Traditional denying, after Aristotle, of ontological status to relations did not help developing a logic of them either.
It should also be noted that the translation of natural language into modern predicate calculus is generally considered artificial, see What are the advantages of Aristotle's term logic over predicate logic?, so it is disputable that "speaking involves relational predicates", at least if predicates are taken as functions on a domain of discourse as in the predicate calculus. So predicate calculus could not be read off of the natural reasoning like syllogistic could be, and alternative resources of natural language were, in fact, used for relational reasoning. A calculus for them has been developed more recently, see e.g. van Benthem, Natural Logic and Ben-Yami, Logic & Natural Language, ch. 6:
"In natural language, pluralities are introduced and specified by means of plural referring expressions; in the predicate calculus, a plurality, which is unspecified by the sentence, is introduced by presupposing a domain of discourse... In the predicate calculus, quantifiers specify how many particulars from a presupposed domain have a certain property; the quantifier in natural language, by contrast, specifies how many particulars of a plurality introduced by a general term have a certain property... The predicate calculus cannot even be seen as a simplified model of a fragment of natural language."
These devices have early traditional precursors, see Hodges, Traditional Logic, Modern Logic and Natural Language. E.g. Alexander of Aphrodisias and Ibn-Sina converted binary relational inferences into syllogisms by changing the domain of discourse to pairs. Other examples of "non-syllogistic inferences" were also discussed by Islamic scholars, scholastics, Leibniz and others, but only ad hoc, see Medieval Theories of Relations and Relational Syllogisms and the History of Arabic Logic. Ockham and Buridan present oblique syllogisms like "Every horse is black, you have a horse in the stable, so you have something black in the stable" as instances of applying dictum de omni et nullo. Wrote Leibniz in New Essays on Human Understanding:
"It should also be realized that there are valid non-syllogistic inferences which cannot be rigorously demonstrated in any syllogism unless the terms are changed a little, and this altering of the terms is the non-syllogistic inference. There are several of these, including arguments from the direct to the oblique — e.g. 'If Jesus Christ is God, then the mother of Jesus Christ is the mother of God'. And again, the argument-form which some good logicians have called relation-conversion, as illustrated by the inference: 'If David is the father of Solomon, then certainly Solomon is the son of David'".
Still, one would be surprised today by the unwavering faith of Ibn-Sina and many 16-17th century authors in the possibility of reducing Euclid's reasoning to syllogisms, De Risi, Leibniz on the Parallel Postulate and the Foundations of Geometry, 3.1 gives a nice review:
to present all geometrical proofs (including those of axioms) as logical arguments
in forma or chains of syllogisms (or other logical inferences) starting from the definitions.
This kind of reduction of Euclid to syllogistic reasoning had already been
attempted, it is true, in the past. The celebrated Analyseis Euclideae (1566) by the
mathematician Christian Herlinus and his pupil Konrad Dasypodius had displayed in
fact the first six books of the Elements as chains of syllogisms (or other propositional
inference rules); and Clavius himself had quoted with praise the Analyseis in his
commentary, reproducing Herlinus’ logical proof of Elements I, 1 and stating that a
similar presentation of the subject could be attained for the whole of mathematics".
From the modern perspective, Euclidean geometry essentially involves multi-place relations like incidence, betweenness and congruence. In practice, "chains of syllogisms" had to be supplemented by inferences from diagrams to make up for the inevitable gaps. This was noticed by Kant, and led to his idea that geometric reasoning is "synthetic" rather than "analytic", i.e. not purely logical. Hindsight is indeed 20/20.
"Nevertheless, it is
true that, even in his most daring geometrical constructions, Leibniz remained somehow entangled
in the classical views, and he wavers between old and new concepts of geometry. Moreover, he saw
the necessity of a logical treatment of relations, but lingered in an improved syllogistics that fell
short of it. The most complete and historically accurate treatment of Leibniz’ theory of relations is