Bubble sort is (provably) the fastest sort available under a very specific circumstance. It originally became well known primarily because it was one of the first algorithms (of any kind) that was rigorously analyzed, and the proof was found that it was optimal under its limited circumstance.
Consider a file stored on a tape drive, and so little random access memory (or such large keys) that you can only load two records into memory at any given time. Rewinding the tape is slow enough that doing random access within the file is generally impractical — if possible, you want to process records sequentially, no more than two at a time.
Back when tape drives were common, and machines with only a few thousand (words|bytes) of RAM (of whatever sort) were common, that was sufficiently realistic to be worth studying. That circumstance is now rare, so studying bubble sort makes little sense at all — but even worse, the circumstance when it’s optimal isn’t taught anyway, so even when/if the right situation arose, almost nobody would realize it.
As far as being the fastest on an extremely small and/or nearly sorted set of data, while that can cover up the weakness of bubble sort (to at least some degree), an insertion sort will essentially always be better for either/both of those.