Standings are pretty much only a formality: all they do is show up on your profile’s trophy list. I agree with the idea of putting the top three somewhere on the tournament page. It may be redundant but certainly a good quality-of-life update. And it removes all confusion.
I think third place in single-elimination tournaments is whoever lost to the first-place winner. I remember as a weak twenty-plus kyu, I lost easily, only to get third place because my opponent went on to win the 4-player single-elimination tournament.