I suppose it all depends on what type of balance you want to achieve. Without knowing much about your systems, I could think about some possibilities:

A) Each attribute is highly independent and function constantly over time. In this case, you have to analyze your tracks. As long as they are well built (in terms of requirement distribution), each attribute will be somewhat balanced. The downside of this is that it increases the relevance of the roll, which doesn't seem to be what you want.

B) Attributes have synergies, or are situational. In this case, some attributes would not strongly influence your performance, but would give you a big boost at some points (e.g. determination might give a big boost towards the end of the race or something to that effect, or sprinting boosts your strength during certain points). This way, you would have core and support attributes, and their combinations would determine the player's optimal strategy.

Unfortunately, I'm more of a tinkerer to provide you with elegant Math to calculate optimally those things. However, I'd include the dice roll in your model and minimize its influence iteratively until you reach a level you find fun, or that it doesn't feel too luck based. You can also avoid people investing in a single attribute if you provide diminishing returns (something in the e^(1/x) function family). You might also try running multicriteria optimization and adjust your variables until the problem is not dominated by only a few attributes.

Hope this helps a bit