In HTLC, after all the parties have locked their funds in the contract, claiming payment happens in a reverse way, starting with the receiver who reveals the preimage of the corresponding payment hash.
What would be the problem if it had started with the sender? After the receiver confirms receipt of the contract, it sends an acknowledgment to sender. Sender releases preimage, timelock set in increasing order across the path, contract initiated by sender having the least locktime. Receiver is the last node to settle the payment.
For reference, I am adding a picture so that one can visualize what I am trying to say.