I don't believe you're correct, but your re-statement of the condition is even more obscure.
The verb is only mentioned once, so it must apply to both conditions. There are two conditions, because there is an or and the and is part of "both B and C" — the both links B and C together.
The statement can't actually be reduced any more, but let's say it's
If they don't have A or both B and C, then X.
- If they don't have A, then X
- If they don't have both B and C, then X
The way to prevent X is to have either A or both B and C together. A real-world example is needing identification: you can get a discount with either a photocard driving licence (showing name, address and photo) or both a passport (name and photo) and a utility bill (name and address).
It would certainly be easier to say "X unless they have A or both B and C."