It turns out that I can make my car do the delayed input thing as well as seen in pgagoobers video earlier, I was just doing it wrong before. However, I found something interesting. Turn telemetry on. Most cars will have some front toe in, so when you're going in a straight line you can see the forces on both the front contact patches pointing inwards. You can play with the steering while watching this, and if you use small movements you can identify the range in the middle of the steering where there's no change in trajectory because of the toe angle. This is what front toe is meant to do, it stabilises the car by requiring larger steering angles to deviate from straight.
You can also see while doing this that there seems to be very little lag between steering inputs, and the reaction of the telemetry. The car doesn't move if you make small steering movements, as it shouldn't. But the telemetry reacts instantly.
Before the tin foil hat brigade points out that they've fudged the telemetry, it'd be a lot hard to make a fudged telemetry system than it would be to just make a decent physics system in the first place. I suspect what we're seeing is a real effect of the physics.
Now, the obvious test after that is to see if toe out makes the car more responsive (as real life physics would lead us to believe). Put your front toe all the way down to zero, a major toe out setting. You'll now see the forces on the front wheels pointing out when you go straight. Do it with the back toe too if you want super responsiveness. Now try and turn. Feels faster, right? It feels faster to me. It's not immediate, but it's not in real life either. It takes time for the wheels to turn, for the suspension to load up, for the tyres to bite and travel. You're never going to get instant response. What we're looking for is a response time that is suitable for a sports car.
So, temporary solution, add toe out. That's the same solution you'd prescribe for a real car which had sluggish handling, which is promising.