My experience relates to dealing with LV Insurance in the UK, but it’s a current ongoing claim so may perhaps be helpful to note…
We had a small but long-running leak under the kitchen sink which went unnoticed until the plywood under the kitchen floor started to lift up! It was caused by a badly-fitted “Quooker” tap (A thing that heats water just for the sink) - there was a bend in an overflow pipe which meant the overflow got blocked and the water dribbled down the wall behind the sink and onto the floor.
Rather than wait for the insurers to send someone out, I contacted a company that specialise in repairing water damage in properties, and they came and gave a quote (around £3000), which the insurance company accepted straight away without requiring their own inspection by an assessor, or even alternate quotes. A cheque arrived from LV within a fortnight.
We’ve had their builder round today as it happens and he has removed the base of the sink unit, stripped out the damaged floor, and put in the dehumidifier and a temporary floor - reinstatement should happen in 10-12 days’ time when it’s all dried out.
So all good so far. It may be that we’ve had an easy time because it was a relatively small-scale problem, but I hope that encourages you in your dealings with your insurers! I’ve met some awkward ones in my time but LV have been great!