A contract certainly does exist between Royal Mail and the vendor (person posting the item). However, the terms of that contract are limited in respect of uninsured loss. I seem to recall ordinary post it a maximum of £37 or thereabouts; the amount of compensation being set out in Royal Mail's terms and conditions for each type of post-service which form part of the contract of service with the vendor (person posting the item).
Most definitely Royal Mail will need documentary proof in order to support any claim for loss and this is usually with any insurance claim including (say) a motoring accident or household loss. Besides proof of posting (and as Going Spare quite rightly points out) documentation robustly demonstrating value would be required. Anyone who believes otherwise would be naïve.
Ultimate if I post 'normal post or signed-for I expect to cover any loss; hence using Special Delivery for important items as this carries a specific insurance for loss.
As an aside in law you can be 'served' documents (say a summons) by regular post; courts accepting simple proof of posting as being sufficient. However, if you have to 'notify' somebody by post (which entails a positive obligation to ensure receipt) then only Special Delivery will suffice as it provides a signature by the addressee (compared to signed-for could be signed by the reception of a multi-occupancy building on behalf of the addressee but not actually the addressee).