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Rather than hide under the table when the Centre for Policy Studies report is mentioned, I suggest that you try reading it. You will then be able to reply that it is a shoddy piece of work, peddling its authors agenda, and not an accurate reflection of the true state of the legal profession, or its fees in the UK (or even England).
The problem with fixed fees in litigation is entirely different. They do not control how much the lawyer can be paid (or ask to be paid) by his or her client. They control how much (or rather how little) a successful litigant can recover from the loser. In so far as the the sum is less than the true market worth of the work properly done for the successful litigant they act as a disincentive to pursue, or defend, genuine claims, and are a barrier to justice.