you are describing GOFAI (good old fashioned AI) which was using things like logical production rules. The issue is that the ability to model the complexity of things like human languages and continuous/real-valued systems makes it infeasible to use at scale. For something like the alice brothers/sisters example, that is a canonical example where GOFAI and/or logical reasoning with languages like prolog excel. A common example in prolog is:
philosophers => human
human => mortal
plato => philosopher
plato mortal? Yes
But encoding enough semantic information to create compelling AI with this type of system is difficult. Some have tried to enter thousands/millions of rules and still the system isn't convincing.
The main breakthrough that has enabled LLMs is an encoding of words that relies on their frequency in being near other words in the english language (using all the text available on the internet). Therefore words like "philosopher" and "plato" become associated in a high-dimensional space (so instead of "plato" you have a "token" with thousands of numbers associated with it).
You can then perform numeric operations on these numbers to come to conclusions. For example, we would expect something like a "human name" to emerge in this embedding space where we could determine if something "is used like a name" in various contexts by applying some non-linear transformations of the word vector / token.
LLMs have simply make a sequence of these transforms, while using prior words it generates as additional input (which allows it to construct sentences). So it is quite different from traditional reasoning. It is better at "fuzzy reasoning" but also worse in situations that require precise results (in fact, at each step it chooses one of a few best possible words based on its stats at random, the variation in this is called 'temperature').