To your point about OCR: I think you'll find that the existing OCR tools will not know where to begin with the 18th century Mexican medical text in the second case study. If you can find one that is able to transcribe that lettering, please do let me know because it would be incredibly useful.
Speaking entirely for myself here, a pretty significant part of what professional historians do is to take a ton of photos of hard-to-read archival documents, then slowly puzzle them out after the fact - not by using any OCR tool (because none of them that I'm aware of are good enough to deal with difficult paleography) but the old fashioned way, by printing them out, finding individual letters or words that are readable, and then going from there. It's tedious work and it requires at least a few days of training to get the hang of.
If anyone wants to get a sense of what this paleography actually looks like, this is something I wrote about back in 2013 when I was in grad school - https://resobscura.blogspot.com/2013/07/why-does-s-look-like...
For those looking for a specific example of an intermediate-difficulty level manuscript in English, that post shows a manuscript of the John Donne poem "A Triple Fool" which gives a sense of a typical 17th century paleography challenge that GPT-4o is able to transcribe (and which, as far as I know, OCR tools can't handle - though please correct me if I'm wrong). The "Sea surgeon" manuscript below it is what I would consider advanced-intermediate and is around the point where GPT-4o, and probably most PhD students in history, gets completely lost.
re: basically perfect, the errors I see are entirely typos which don't change the meaning (descritto instead of descritta, and the like). So yes, not perfect, but not anything which would impact a historical researcher. In terms of existing tools for translation, the state of the art that I was aware of before LLMs is Google Translate, and I think anyone who tries both on the same text can see which works better there.
re: "irrelevant books," there's really no way to make an objective statement about what's relevant and what's not until you actually read something rather than an AI summary. For that reason, in my own work, this is very much about augmenting rather than replacing human labor. The main work begins after this sort of LLM-augmented research. It isn't replaced by it in any way.