That isnt true is it? An accident management company will take their fees from the third party, not from your payout. At least that is my experience of them?
No. Reforms in April 2013 meant solicitors were unable to recover the success fee from the at fault party's insurer.
In the event of a non fault accident without injury and no other losses this is pretty straight forward - no solicitors are required. You simply get a replacement vehicle or the money that it's worth if it's written off. Accident management companies manage this and claim their costs through extortionate hire prices and then invoice the at fault party's insurer for £200/day for hiring a Corsa (some exaggeration in the cost there).
In the event of a non fault accident
with injury
and/or other losses a solicitor will likely be involved, and they must recover their costs from your motor insurance legal cover (this does not effect your no claims on your motor insurance); or your home insurance legal cover (this might effect your no claims on your house insurance); or you agree to pay up to 25% of any compensation to your solicitor.
The solicitors somehow insure themselves against loss in the event of a no win situation in a no win no fee agreement.
This means solicitors are now not taking on ambiguous whiplash cases - they've started to take on clear cut ones where the third party are clearly at fault (significant rear end damage etc).
I'm no expert in the matter but that's the current state of affairs as far as I'm aware.

It was all meant to clamp down on those silly ambulance chasers which drive up insurance costs.
I used to advise against legal cover - but for the sake of an extra £20 to the premium, and change in law, I now advise for legal cover.