As long as they are known in advance and don't change, I'm happy with any (sensible) rules.
Geoff's suggestion has a few small knock-on effects though:
a) It favours coaches who own several teams. This requires more financial investment. You don't just need to own 3, you need to own 3 of the 6 or 7 "power" races.
b) It favours coaches who go to more tournaments (more so even than now). At the moment you need to play 3 tournaments to qualify. Placing high in 3 tournaments with 3 different teams is harder than placing high in 3 of 8 tournaments with (say) 4 different teams.
c) If favours coaches who do well at tournaments early in the year, e.g. Poo Bowl, Watermonkey. Everyone has a favourite race that they expect to do better with. As a purely hypothetical example, some coaches do well with powergaming teams like Amazons, but dreadfully with High Elves or Khemri.

Get a good score with your favourite race early on and you have "one in the bank" and can start experimenting with other races.
But overall I like what Geoff is getting at. Not sure it is perfect in execution yet. Maybe saying that a coach can't have
all three results with the same race would be in the middle ground?
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As an aside, how about a UK Stunty Champion for next year? Same system as this, except that only Stunty teams count.

*This post may have been made without the use of a hat.