Why are insurance claims handlers so bad at handling claims?

For the first time (as far as I can recall) in our 6yr history we have recently had cause to claim on our public liability insurance. We fitted a ball valve to a storage tank, the valve subsequently failed, the overflow couldn't cope and water flooded into the customer's property, damaging paintwork and carpets.

No huge drama, but large enough cost (well into four figures) to warrant claiming on the insurance. How long would you think it might take, from notifying the insurance company to them authorising the repairs (bearing in mind this is water damage, so pretty unpleasant to live with for any length of time)? A week? Two weeks?

Six weeks on, and we still haven't had any decision from the insurers (AXA, who have subcontracted the claims handling to Cunningham Lindsey). In the meantime, we've just gone ahead and repaired everything that we can do ourselves, but customer is still left with old carpets. We don't want to tell him to just get new carpets and send us the bill, because the insurers might want to inspect the old carpets.

Why does it take so long? Why can't it work like this:

Day 1: Notify insurance company of potential claim
Day 1: Insurance company contacts customer and arranges to inspect damage (in, say, 5 days time)
Day 2: Customer gets estimates to repair damage
Day 6: Insurance co inspects damage, reviews estimates
Day 7: Insurance co authorises repairs

Why does it take weeks and weeks? And why do we have to manage the process (i.e. we only see any progress at all if we badger the insurance company)?

Although this is the first time we've had to claim on our own insurance, we've often had to claim on other peoples for road traffic prangs, and the process is always extremely painful even when there is no dispute about liablity.