Assuming those cars have a blackboxed drive log, it should be easier to determine what specifically is at fault and where.
Just like now, you'll still have to carry insurance, just in case your car is the one that screws up or you want coverage for others' faults. You'll still want to read that contract to verify that it covers what you want so that you aren't waiting for a lawsuit to resolve.
If I'm in an accident, my insurance company writes me check if I'm covered. If it's my fault (within certain limits) I still get medical and mechanical paid, because I pay them for that coverage. I have Un/under-insured coverage, because the minimum isn't really adequate. I have road hazard insurance. Etc... Someday I'll probably have Automated Driving System Failure coverage.
But wrecks should be MUCH less common. Comprehensive insurance policies will get cheaper (right, I'm an optimist) to encourage more expensive coverages that guard against stuff that doesn't happen that often.