Logicism's original goal certainly was not to diffuse Platonist impulses, although it was later adapted to that end, Frege was a devout Platonist. It was an epistemological reduction programme (aside from the more technical mathematical project): show that mathematics reduces to logic, which is more "secure" on any conception, and deal with the justificatory status of logic later. Frege, Russell, early Wittgenstein and Carnap had different ideas how, from platonism to positivism, and how much can be reduced, from primitive recursive arithmetic to all of mathematics. What attracted positivists to logicism was the hope of eliminating platonist metaphysics by reducing mathematics to conventions, because it seemed initially plausible that logic could be so reduced (this hope was later dashed by Quine's Truth by Convention). See Friedman's Logical Truth and Analyticity on the evolution of logicism from Frege to Carnap until the final blow to it was dealt by Gödel's incompleteness.
As for "logical objects", like connectives and predicates, they are eliminable. Wittgenstein in the Tractatus came up with the idea of combinatorial interpretation of connectives, they then stand for no objects at all, "logic takes care of itself" (of course, late Wittgenstein concluded that all of mathematics is not "about" anything, and is normative grammar in disguise, but that was hardly logicism). But according to Friedman, the Tractarian logicism recovers, at most, only primitive recursive arithmetic.
Quine famously adapted Russell's paraphrase to deny ontological existence to predicates. His criterion of ontological commitment is "to be is to be a value of the bound variable", i.e. to be is to be in the range of existential quantifier of scientific theories after paraphrasing out dispensable fictions, like Pegasus (sets and numbers can not be plausibly paraphrased, according to Quine, they are indispensable). He then replaced Frege's second order logic by first order logic with "semantic ascent" that does not quantify over predicates. Instead, we move to the meta-language and "paraphrase" second order quantification over predicates with schemata that contain placeholders fillable by definable predicates only. This is a formal mirror reflection of medieval nominalism, "real" common natures (objective predicates) are replaced with nomina ("words" for symbolic predicates). Of course, Quine was no logicist, but one could say that he picked up the pieces from Carnap's version of it that could be salvaged by embedding logic and mathematics into his holist web. In Epistemology Naturalized he admits that the original promises of logicism ring hollow:
"Mathematics reduces only to set theory and not to logic proper. Such reduction still enhances clarity, but only because of the interrelations that emerge and not because the end terms of the analysis are clearer than others. As for the end truths, the axioms of set theory, these have less obviousness and certainty to recommend them than do most of the mathematical theorems that we would derive from them. Moreover, we know from Gödel's work that no consistent axiom system can cover mathematics even when we renounce self-evidence. Reduction in the foundations of mathematics remains mathematically and philosophically fascinating, but it does not do what the epistemologist
would like of it: it does not reveal the ground of mathematical knowledge, it does not show how mathematical certainty is possible."
Friedman, for his part, argued that Quine's "web of belief" is not sufficiently stratified, and that mathematics and logic enjoy certain autonomy (his own student Parsons and later Maddy expressed similar sentiments). This opens up a possibility that the reduction to logic and set theory may be more meaningful epistemologically than mere pragmatic convenience. And more recently there is a movement that is even closer aligned with the original logicism, the neologicism or neo-Fregeanism:
"The neo-Fregean movement seeks to reveal that a significant amount of mathematics is analytic. This is a stronger claim than that it is a priori and derives no part of its justification from empirical science, or even from successful applications within the empirical sciences. For that would hold of mathematics (or indeed any other branch of knowledge) conceived of as synthetic a priori. The neo-Fregean maintains in addition that significant parts of mathematics flow logically from principles that are analytic of (or definitional of) their central concepts or predicates, such as ‘natural number’ or ‘real number’. That is, they flow from the very meanings of those central predicates."
Of course, neo-Fregeans learned Frege's lesson, and admit that logic does not express indubitable "laws of thought", and that analytic "truths" can be discarded on pragmatic grounds, albeit not empirical, but that is still much closer to Frege and Carnap than to Quine. Not all neo-Fregeans are prepared to go as far as the above SEP quote, however. Heck in The Julius Caesar Objection states more modest goals:
"Not even the claim that numbers are objects is required for Frege’s proofs of the axioms of arithmetic. What is required is that expressions of the form
“the number of numbers less than 5” should be of the same logical Sort as those of the form “the number of Roman emperors”... The attractions of the genetic story told at the beginning of this section do not depend upon the claim that the various instances of Hume’s Principle are logical truths, analytic truths, or any such thing. Frege’s most fundamental thought—that our knowledge of the truths of arithmetic derives, in some sense, from our knowledge of Hume’s Principle—could well be true, even if it does not have the epistemological implications he had hoped it would."
