I know this game has tons of optional logs within it to unlock and fill out and it seems like a lot, but I honestly quite enjoy having them and I think it would be nice if we had logs to cover story tidbits and overworld mobs.
A lore log could work as thus:
The log could include all sorts of info such as locations, cultures, history, magic or aether-related stuff, science, major and minor characters, etc.
Every time you learn something new about the world in a quest, it's added to your lore log.
Every time you meet a new character, they're recorded in a "people" section of your log.
Every time you encounter new lore you'll get an "entry added to log" text box. This could show up at the end of a quest, or after interacting with stuff like the books in Gubal.
Large entries will be gradually expanded upon with each side quest or duty you complete that contains relevant information.
Every time you get a new or updated entry, there's a little green notification on your log menu icon that directs you to the updated section.
Characters can be divided into main story, job story, and major side content sections, and minor or side quest characters can be divided by location.
At the very end it could include a glossary of terms that you can type in a search bar for, and selecting a term pull up entries with that term highlighted.
There could also be lore entry drops like field notes to be gotten from duties. Essentially, this idea is kinda like an expanded version of the notes from the Nier raids but for our lore. The only downside is that it would likely have to be filled out retroactively, requiring people at endgame who want to complete the log to redo old duties, but I personally wouldn't mind that much. They love giving us reasons to run old content, after all, and I'd be willing to run through old dungeons if it let me compile world lore.
I know some people would probably think "well wouldn't that discourage people from buying the Encyclopaedias?" but the difference is that the ingame log would have to be built up, while the encyclopedias already have everything. And since the production of a new volume would be very difficult for Square, this could be a good way for them to appeal to lore nerds without having to do that in order to compile post-4.3 lore.