Yes, that's how the Tournament race mode works in ARC One & ARC Air. Like you, I remember being puzzled the first time I used Tournament with more than three racers. Pen & paper (or a spreadsheet) does the job, but it would be nice to have the option in the ARC app for everyone to race each other at least once in Tournament mode.
I haven't used Tournament in ARC Pro. I suspect that with up to six cars on track at the same time, it will work differently?