To understand the relation between Leibniz's theory, on the one hand, and modern ϵ-δ definitions and constructions of the continuum, on the other, it is helpful to make the following distinction. The distinction is between the following two items.
(1) On the one hand, we have the procedures employed by Leibniz and other pioneers of the calculus, and their procedural moves in deriving various new results such as Leibniz's derivation of the product formula for differentiation, and Euler's various solutions of the Basel problem, to give some elementary examples. These procedures can be characterized as syntactic material because at this stage in the development of mathematics, its practitioners were not overly concerned with semantic issues (see below).
(2) On the other hand, we have the issue of the ontology of mathematical objects, i.e., what are the basic objects mathematics uses, how one axiomatizes their behavior and justifies their "existence" relative to appropriate foundational theories. These semantic issues only began to emerge starting around 1870, and resulted in the modern foundations for analysis in particular and much of modern mathematics in general.
To answer your question, I would say that the contributions of Leibniz and others would today be viewed as centering on the syntactic side of the mathematical endeavor. They developed the calculus in the sense that they understood its procedures and syntax, which remain today largely as they have been developed by Leibniz, Euler, Lagrange, and others.
The semantic side of the story did not emerge until much later. This observation is sometimes expressed by claiming that the pioneers of the calculus were not "rigorous" but this I believe is a misleading statement: Gauss and Dirichlet similarly did not have access to modern sematic theories and in this sense would also not be "rigorous", but in fact there are almost no errors in their work. For this reason it is more meaningful to acknowledge the syntactic contribution of the pioneers of the calculus than to apply the essentially anachronistic straitjacket of "mathematical rigor" in analyzing their work.