You have diagnosed it correctly: the problem is earlier than the repeating. You are not forgetting the name, you never encoded it, and those feel identical from the inside.
The reason is that the moment someone says their name is also the moment you are managing a handshake, deciding what to say, and running whatever self-consciousness you brought. The name arrives during the busiest two seconds of the interaction and gets no attention at all.
So the drill is about buying attention back:
Decide in advance that the name is the only thing you are doing. Before the event, not during. Give up on having a good opening line ready — you cannot do both, and the name is worth more.
Repeat it back with a pause, not inside your greeting. "Sarah" — stop — "good to meet you." The pause is what does the work. Saying it as part of a fluent sentence is exactly as forgettable as hearing it.
Use it once more within the first minute, in a real sentence, not as a tag. That second retrieval is worth more than the first repetition, because you are practising recall rather than repetition, and recall is what you will need later.
Use it when you leave. Third retrieval, spaced by the length of the conversation.
That is spaced repetition compressed into three minutes, and it is the difference for me between six names lost and five out of six kept.