One potentially controversial answer might be to say that Russell adopts a position that problematizes the notion of ontology. Russell is sometimes read as a Structural Realist (e.g. William Clement, 1953), in that while he takes the world to be made up of parts, they are realized in the world as a whole and accessed through the structural similarity between thought and fact given by Logic. Logic is what splits the world up into manageable and malleable chunks of data, rather than things coming as pre-packaged individuals.
Part of what lets him do this is that he's working on the foundations of his earlier work in mathematical logic given in his Principia Mathematica (Wiki article). There, Russell attempted to consolidate earlier work by Gottlob Frege to yield definitions for mathematical classes from an axiomatization of a restricted, higher-order logic framework. Class/set discourse is very expressively useful in both analytic philosophy and mathematical model building, and showing that these "fictions" can be reduced to fairly simple assumptions of logical relations or Types of symbols in our language means that logical work has a much broader scope of application. (The literature on this general project is enormous, but the SEP article on "Conceptions of Analysis" might be helpful as an intro!)
Russell in PoLA thinks that as long as languages (any suitably grammatically integral languages, not just those adopted by professional mathematicians) have a properly typed logical grammar where we distinguish elements of interest as "particulars", we can use the most useful bits of set theory and mathematical/philosophical analysis without requiring that we have abstract mathematical "objects" to refer to. Moreover, once we do have this, we'll be able to find that we generally need far fewer particulars than we first thought to account for all of the facts we want to account for, because what counts as an individual hinges on logical theory rather than surface syntactic form. Lecture 7 looks at some of this in more detail, but the following quote seems particularly apt:
The theory of types is really a theory of symbols, not of things. In a proper logical language it would be perfectly obvious. The trouble that there is arises from our inveterate habit of trying to name what cannot be named. If we had a proper logical language, we should not be tempted to do that. Strictly speaking, only particulars can be named. In that sense in which there are particulars, you cannot say either truly or falsely that there is anything else. The word "there is" is a word having "systematic ambiguity", i.e. having a strictly infinite number of different meanings which it is important to distinguish.
Propositional functions seem to be understood in terms of formal relationships between the different parts of a properly logical symbolic language. The problem we might have now is as to what, strictly speaking, we ought to say about the foundations of proper logical structure and the idea of a logical particle, and you're right to point out that without some sort of model theory, Propositional functions seem completely mysterious. There doesn't seem to be any particular consensus on what Russell had in mind in terms of what they really are (see the SEP article on this issue).
But reading Russell as a Structural Realist may give some clues as to how someone trying to renew his philosophical project might be able to think about the nature of propositions and propositional functions. Russell's Principia stipulates what logical statements there are and how they relate through his axiom systems. A Structural Realist might say that when we engage with the world, what we are engaging with is its axiomatically systemic nature, which we take to be "the same as" our logical language. We build and work with axiomatic theories of the propositional structure of the world by relying on Principia's specifications of logical structures. And as such, ontology is strictly subordinate to logical specification.
Actually, structures are "out there" as realized in the empirically observed world too (in as much as the idea of an "out there" makes sense in Russell's metaphysics; it would be probably more technically accurate but much less charitable to say the structure is "just there"), but in order to further evaluate some structure as an object, we would need to logically situate it inside another axiomatic theory to give us its objects and functions. (This can in principle be done, since Type theory is hierarchical, but the behaviour of higher types or a type/category of "fictions" isn't very well specified by Russell's theory) The question of "what we say" about propositional functions and individual bare particulars beyond what prior specified logical axioms can tell us would be thus easily answered: "nothing at all". (The structuralist strategy and some consequences of it for maths received much more attention after Paul Benacerraf's arguments in "What Numbers could not be" (1965))
If this is what we're committed to, we can still get propositions and propositional functions out of it at the end, because that's how logic seems to work, but their identities are in some sense parasitic on how the world is "organized" into facts. Had the world been radically different, with more complicated types than we currently think there are, the identities of propositions and propositional functions might also have been different. That may be a step too far for anyone looking for logical structure to serve as a stable foundation.
(all quote marks are scare quotes, and not necessarily Russell's own terminology.)