The LLM is still a "normal algorithm", just one with a fairly large dataset to use. Assigning the algorithm part of LLMs magical properties hinders understanding. The work needed to pick the next output token is very much a classical algorithm.
(Now, the weights used, those we kinda really don't understand the same way we understand the processing, and the approach to looking for structures in weights sometimes looks more like archeology or anthropology than computer science.)
It sounds like you're trying to express some kind of "but LLMs are so much more" thought. Yes, very much, they are. It's because of the size of the data, there's interesting emergence there. They're still a normal algorithm. (And our brains aren't quite like that; biological things are much more random/chaotic and generally non-reproducible. And the data and algorithm aren't separate.)
Algorithms can be based on training and/or use data just fine, too. https://arxiv.org/abs/1712.01208
(Now, the weights used, those we kinda really don't understand the same way we understand the processing, and the approach to looking for structures in weights sometimes looks more like archeology or anthropology than computer science.)
It sounds like you're trying to express some kind of "but LLMs are so much more" thought. Yes, very much, they are. It's because of the size of the data, there's interesting emergence there. They're still a normal algorithm. (And our brains aren't quite like that; biological things are much more random/chaotic and generally non-reproducible. And the data and algorithm aren't separate.)