Names are gone before the handshake finishes — is there an actual drill for this, or do people just pretend?

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.

Adding the part that makes it stick past the same evening, which is a different problem from the one above.

A name is an arbitrary sound attached to a person. Nothing about "Ravi" follows from the face, which is why it has nothing to hang on. So give it something to hang on, and make it something you can see, not something you think.

What works for me is finding one visible feature and linking it, however stupid the link is. Stupid is better — absurd links are more memorable, and nobody hears them. The rule is that it must be a feature you will see again next time. A hairstyle changes; the shape of a face does not. Clothes are useless.

Two things people do that fail:

Linking to the context instead of the person. "The woman from the finance table" gets you nothing when you meet her in a corridor.

Trying to remember the name as a sound. It is the least memorable form. Attached to an image it survives; on its own it evaporates.

And for what it is worth on your last question: no, most people are not faking it, but a lot of people are quietly avoiding the moment — never introducing two people to each other, saying "you two know each other?", using "mate" and "you". If you have been doing that, it has also been preventing you from ever practising, which is its own trap.