Only Lab-based tests?
Not when Mythbusters went about it. They have compared Drunk driving, Mobile use (with and without hands-free) and tired driving. None of them compared well with the 'normal' state of the drivers.
You may not feel that they used a sufficiently large sample size but the differences they recorded were big enough for each driver to be relevant and possibly a lot more relevant than stats which start by removing lots data to eliminate 'other' factors. You say that even if most ignore a ban at least some will observe it but could those be the same people that were careful before the ban and did not have accidents whilst on the phone?
Gets us into the territory where they list the cause of an accident as 'Speed'...this is simplification of cause for ease of collation. Speed is the cause of every automotive accident as without it no two object can come into contact with each other (although technically 0 is also a speed). The critical word missing is 'inappropriate'. Go at 60 round a bend marked at 30 and the likelihood is you will have an unintended occurence, travel at 90 on an empty dry well-lit motorway way and your are much less likely. Use your mobile phone in the midst of rush-hour traffic to get directions when you are lost and it is when, not if, you will hit something. Do the same on that aforementioned motorway and the reduction of attention on your actual driving is not so relevant.