A very short question:
If, following Carnap, we can represent a possible world as a state description starting from a quantified language L, why does it make sense, following standard modal logic, to take worlds as primitive instead of just a set of sentences in our language?
It was David Lewis who advocated to take all possible worlds as primitive ontological commitment instead of just a set of sentences in our language within our actual world known as Modal realism.
Lewis gave a variety of arguments for this position. He argued that just as the reality of atoms is demonstrated by their explanatory power in physics, so too are possible worlds justified by their explanatory power in philosophy. He also argued that possible worlds must be real because they are simply "ways things could have been" and nobody doubts that such things exist. Finally, he argued that they could not be reduced to more "ontologically respectable" entities such as maximally consistent sets of propositions without rendering theories of modality circular. (He referred to these theories as "ersatz modal realism" which try to get the benefits of possible worlds semantics "on the cheap".)
So your intuitional position is called argument from ways which defines possible worlds as merely "ways how things could have been" and often used as non-concrete "ersatz modal realism" to argue against Lewis's modal realism. But if you just use a set of propositional sentences as a replacement, these sentences are nothing but representations which are not ontic concrete objects themselves. As representations, they must be further derived by some distinct ontic object in our actual world, but then how can you make sure your imagined set will exhaustively contain all representations there are? Maybe tomorrow some aliens are suddenly known in our actual world but your original set definitely excludes this case! In contrast, possible worlds are ontic concrete and thus aren't exhaustively defined by some representational set of sentences. This is the main advantage imho for PW realism which was elaborated in his third chapter of On the Plurality of Worlds.
Another argument for taking worlds as primitive instead of taking modality (necessity, possibility, etc) as primitive lies in the fact that referenced here:
But modality is mysterious. Modal properties do not fit easily into an empiricist worldview... Modal properties do not seem to stand alongside fundamental qualitative properties a part of the furniture of the world. Thus, modality cries out for explanation in non-modal terms. Theories that take modality as primitive, then, will sacrifice much or all of the explanatory power of modal realism.
Perhaps you are misinterepting the meaning of 'possible world', in carnaps sense of a state-description.
He defines a state-description in his book: The Logical foundations of Probability. :-"A sentence (or class of sentences) describing completely a possible state of affairs of the the universe of discourse/ a "conjunction" (or class of sentences) containing as components (or elements) one sentence out of each basic pair".
Where the -Basic Pair, is defined on an atomic sentence and it's negation. Therefore, a state-description is known within 4 conditional states:
There's a range within the state-description defined on pre-defined logic set-out by Carnap, but also a Q-property which defines only those properties in the language L, that takes the binomial factor $2^n$, given the definition of a basic pair.
The primitive predicates are known as containing signs, attributes and their relations. Therefore, in detail to your question about taking primitive predicates over sentences in the language L.
Primitive predicates set out to define the range of individual constants (which Carnap defines as events) over the basic pair. Therefore, a universe containing both L-determinate sentence and L-factual sentences, hence the primitive predicates are only blocks for defining the tautology of the universe. Whereas, the sentences within L which at times are defined on the sentential matrix:- {i}(M) which contains only sentences with free-variables (I believe they're quantifier free),although this corresponds to what he calls molecular predicates, which are predicates introduced as an abbreviation for a molecular predicate expression. This expression consists of both primitive predicate and connectives (Carnaps example: '~P1 V P2').
It is not necessarily the case that a world can be described by a set of sentences. The set of all sentences expressible in a finite alphabet is countable. Therefore, the set of sets of sentences has cardinality equal to the continuum (the set of real numbers). What if the number of worlds has greater cardinality than the continuum? Then the vast majority of worlds cannot uniquely correspond to a set of sentences.
(And note, the above assumes we may describe worlds with an infinite set of sentences, which itself is questionable, especially if we have no way to compute which sentences fall within the set for a particular world.)
