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> you have to zero out your velocity at exactly the right moment.

Not quite. You can compensate with a landing gear that can survive your terminal speed going to zero very quickly and absorb the extra energy.

The math is much simpler on the Moon because there is no atmosphere.

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There is no terminal velocity on a body with no atmosphere

Show that landing gear that can handle an impact at 1km/s, I'm pretty sure there's a lot of nation states out there that would be interested.

It's less unreasonable than you think. The Ranger 1 spacecraft attempted to land a payload with an impactor before the US had figured out how to land more carefully.

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