But paradoxically, while B is more likely to get to $400 first and thus compete for the car, A has a higher expectation of money winnings.
At a high level, it is because A is much more likely to win by a large margin.
Looking at the 128 possible A/B combinations for all seven rounds, we see that for the 16 cases that $400 is earned in just 4 rounds, it is 4 times more more likely that A will have won (and B got nothing). There is a slight advantage to A in the 32 cases where $400 is reached in the 5th round. While the advantage shifts to B for the cases where $400 is reached in the 6th or 7th round, the advantage is not as great as A had in the other cases.
I can email you a spreadsheet (I had to convince myself) if you would like.