My point is that despite Finland having a perfectly good third person singular for people we usually use the even more general one

The reason for that is because “se” as strictly a “thing” pronoun is artificial “book language”. When standard literary Finnish was being developed in the 19th century, its inventors wanted to have a person/thing distinction in pronouns like the “civilized” languages had, so they arbitrarily assigned “hän” as a person pronoun and “se” as a thing pronoun. That distinction is artificial, and has never stuck in spoken Finnish.

Originally there was a difference between “hän” and “se”, but it was grammatical: se was the general third person pronoun, hän referred back to the speaker (logophoric pronoun). Compare:

• Antti sanoi, että se tulee. (Antti said that someone else will come.)
• Antti sanoi, että hän tulee. (Antti said that he himself will come.)