Your version of radical empiricsm simply cannot work (i.e., that all cognition is explainable by experience), and even the logical positivsts understood this, because they saw that one
must presuppose the analytic/synthetic distinction for the viability of their empiricst project.
The reason is that syntetic sentences, that is, sentences which derive their meaning from experience, must presuppose some analytic definitions to function as 'synthetic' in the first place. Because think in what sense can a sentence said to be 'derivable' from experience: it must mean that there is some general a-priori rule which justifies you in inferring the sentence from a certain experience. Thus the positivists thought that all empirical terms have a fixed meaning or definition (or at least we can construct such definitions for a 'perfect' scientific language). So for example 'cat' is defined via the experience of such and such shapes and colors occurring in a certain configuration; and therefore in the present of such an experience it follows
analytically that you are seeing a cat (because this is what the term means). And this is where their famous verification criteria of meaningfulness comes from: every empirical sentences must be connected to experience (empiricism); to be connected to experience is to be verifiable by experience; and to be verifiable is to have an analytic definition -
prior to any experience - that tells you which experience should count as the verification of the sentence. Otherwise, if you don't have such definitions that fix the meaning of your empirical terms, then nothing could follow from your experience, no sentence could be ever verified by experience (and hence you wouldn't have empiricsm).
The conclusion is therefore that there cannot be only synthetic sentences, but also analytic which are true independently of experience. And this is also how they thought that science works: you have a theory from which (by virtue of the meaning of the terms it contains, as they have been defined) a set of certain possible observations follow. Then when you go on testing the theory, if the predicted observations obtain, then the theory is verified; if not then it is disproved. And the idea here is the dame: unless theory and observation are connected by definitions, no observations could follow (deductively or inductively) from any theory, and there will be no way to test it experimentally, in which case science would be impossible.
And as a historical sidenote, it is important to note that though Quine famously attacked the analytic/synthetic distinction, what he was primarily concerned with is the positivists' conception of the 'meaning' of sentences (this is why he talks so much about synonimity in "two dogmas"); however, he himself didn't reject the distinction (which was central to the positivsts as well) between theory and observation, but only claimed that they are mutually interdependent, and that observation cannot verify or disprove individual sentences, but whole theories (and maybe the whole body of science). And so according to Quine, the way experience is connected to our body of beliefs is not via "definitions" but whole theories or world views ("conceptual schemes"), but it still retains the same idea that our experience is mediated by logical connections which themselves are prior to experience.