Soft spring / dampers + big anti-roll bars

It's quite a complex topic, but to my understanding it essentially comes down to reduced grip and damping difficulties.

Large anti-roll bars can reduce ultimate grip by increasing weight transfer. They're working by trying to lift the inside wheel, not good when the maximum grip will be achieved with the most even loading on the tyres.

They also cause issues with damping, as they're asking the dampers to deal with 2 different spring rates, one in roll and single wheel bump, and a different softer one in full-axle bump and dive/squat. This can make it hard to match the damping rate to the spring rate. This is an issue even if you're not worried about performance, as miss-matched dampers suck for ride quality just as much as performance.

There's also dynamic effects to things like turn-in stability, but I'm a little rusty on those, so I'll let someone else explain them!

Essentially ARB's can be great in moderation as you can get a decent amount of roll-control before their rates get high enough to mess things up noticably, but rely on them too much and the compromises start becoming apparent. As with anything suspension related, it's all a balance between help and hindrance.

On the whole you'll find rally and off road vehicles running little to no anti-roll bars, and when they do it's often for tuning the under/oversteer balance easily rather than to provide any real roll. Fully fledged circuit-racing cars will run what look like stiff bars, but relative to their high spring rates they're actually reasonably soft, and again they're mostly there just as an easy tuning tool for quickly dealing with changes in tracks and conditions. Road cars kind of do exactly what you describe, and use them to kill some roll without having to use spine-jarring spring rates, but there comes a point where it's no longer helping either ride or performance.