The downside of this conversion, at least in my view, is that closing the start gate prematurely can screw up the results for the currently-running race. Even a "well-trained" starter volunteer can find it difficult to resist the temptation to close the gate and start staging the next race before the last car crosses the finish line. The problem is even more pronounced with less experienced volunteers (and experienced volunteers can be hard to come by).
The laser gate uses a more complex system, and isn't susceptible to this issue: the laser has to be explicitly activated, either by using the reset button or under software control. Reverse-engineering the circuit, I realized that adding a single transistor would achieve the same result for the mechanical switch. Here's the circuit:

The green wire is a ground. The "reset/arm" button grounds the yellow wire, which both resets the timer and provides +5v on the black wire. (In the laser gate, this +5v turns on the laser.) In this circuit, +5v on the black wire causes the transistor to conduct, and allows the closed start gate to be detected as a ground on the red wire. Once the race starts (the red wire gets pulled low), the black wire returns to 0v, and the transistor stops conducting; closing the gate now will have no effect until the timer is reset.
My kitchen table prototype seems to work as expected/desired. I've got a race coming up in a few weeks where I hope to try out the "finished" version.
[Edited to add link to a third topic.]