It depends on one’s theory of grammar how much abstract structural grammatical knowledge we assume kids have. There’s a big literature on modeling category learnability. It’s also something of an ideological question now too (one of the splits between formal Chomskian linguistics and other flavors of linguistics, formal and otherwise, is based on what opinion one has about this question.) It’s pretty clear that kids aren’t just copying sound strings though. For example, kids get word boundaries, but the cues for where words start and end in speech are very subtle, and usually not realised at the boundary itself.

My own made-up explanations for this is similar to yours, but appeals to a slightly different knowledge — there are (at least) two ways learners can go wrong. Second language learners get interference from the collocations of their first language, and they analogize (overgeneralize) rules from other collocations. Preposition use in English is a good place to find this sort of overgeneralization because it’s not only somewhat arbitrary, speakers also vary for some (but not all) prepositional uses. For example, “different from”, “different to” and “different than” are all found and are all more or less acceptable to different groups of English speakers, but “similar” can only take “to”, not “from” or “than”, and “other” takes “than”, not “from” or “to”.

The first is similar to what Andrew was arguing: if the verb “help” takes the dative case in my first language I’m more likely to assume that it takes the dative in the second language too, unless I have knowledge to the contrary. (I don’t know to what extent errors fall into direct morphological mapping vs form-semantic mapping; e.g. if Language X marks animate objects with the genitive and inanimate ones with the accusative, and Language Y marks them all with the accusative, will L-X speakers who are L2 speakers of Language Y use their genitive/accusative distinction in Language Y?)

The second pattern — overgeneralization of categories, is also found in child first language acquisition. For example, when English kids acquire past forms, they typically go through a stage of putting “-ed” on everything. Then they acquire some of the minor categories, and over-apply them too (e.g. bring ~ brung by analogy with sing ~ sung). Sometimes these stick into adulthood (dive ~ dove is actually one of these historical overgeneralizations).

Unlike math, language doesn’t even fully come with the option of rederivation. There is an approximate logic to it, but it will not always help to tell “from” and “of” apart. Perhaps it would be approximately correct to say that the difference between the reasoning and the memorization is the difference between the high school spanish class and an immersion program.