Thanks. Yes, I have tried it yesterday with the correct rules, and I think it is much better.
We first played this wrong (As we missed the bit in the end of the race phase section and read it as play only continued to the left when 8 cards weren't out). The problem it causes is with the wrong number of players, you could end up having the same person ending the race phase by playing just 1 card for multiple rounds through bad luck. It's definitely 'fairer' rotating the first player normally.