Should retailer have to prove damage caused by coin?

Upward dents in the drum can only be caused by an object getting trapped between the inner and outer drum. It's an interesting thought you have about them being able to prove anything because if there is no coin to be found then technically there is no "proof".

However, the dents in the drum alone and the damaged tub are proof that something was loose inside the washing machine, almost certainly made of metal. You could argue that without a coin being found there is no proof that it was a coin, and as the onus is on the retailer to prove that it isn't a problem with manufacture (within the first six months) it might be interesting if you decide to challenge them for proof. I would expect them not to back down though as they would be pretty confident that it was caused by a coin. But I have heard of at least one case where a manufacturer insisted that such a similar fault was caused by a coin but the customer found a screw, which was subsequently admitted to be a screw used in production by the manufacturer.

In the industry it is extremely common to find coins left inside pockets which subsequently damage the washing machine. It's also not that unusual for no coin to be found at the scene. I agree that it is a shame that modern washing machines can't stand up to a coin without being potentially destroyed. This never used to be the case when they used metal outer drums. However, at the end of the day coins left in pockets is negligence by the user and there's no getting away from that.