So if you use such chain for multiple payments and one participant in the chain closes prematurely, it is as if no payment has happened? Or is anything preventing this scenario?
No. Each send acts individually. If you make 3 payments and are in the middle of a fourth when one party in the chain closes a channel used in the payment, then only the fourth payment does not go through. The rest remain untouched. Note that this scenario can only happen if the preimages for the HTLCs are not available to everyone which means that the payment is not yet complete.