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[More of an extended comment.]

I think I disagree with the premise of your first paragraph... That being said, while I don't know of a database, there has been some work systematically studying small Turing machines. See Small universal monotone Turing machines, https://cstheory.stackexchange.com/a/20980/129, https://cstheory.stackexchange.com/a/11614/129. While some of this is interesting, I would say it has yet to "bear much fruit (i.e., understanding)." I hope someday it will.

There's also a mismatch between listing small Turing machines and understanding algorithms that are built by people (much as there is still a gap between listing all genes and understanding what they do, though not quite the same). Namely, an ontology of algorithms that are built by people should including things like those mentioned in Thomas Klimpel's comment. (Note: it should also include holographic algorithms, a relatively recent surprise in the ontology of algorithms!) But an ontology of small TMs looks completely different. Given how small a corner of the space of algorithms people have explored, and given the uncomputable problems involved, I highly doubt there would ever be a "complete" ontology of algorithms - either in the sense of small TMs or in the sense of all algorithms people would build. But I suppose having a partial one might still be interesting for some purposes...