Here's an example. You updated the code that interfaced with the database. But you forgot to update the stored procedures within the database. As a result, the database is now being put in an inconsistent state with every transaction that uses your new code. That is the kind of thing that a software design doc can help you remember, because it is supposed to make you think through all the stuff.
And if you're going to say "Your business stuff shouldn't be able to get into an inconsistent state", well, there's a lot of businesses that have potential landmines laying around. You can say they shouldn't. You're right, in an ideal world. But in this world, they do, and you have to live and work in the world that we have.
Now, in fairness, a good AI check might turn up that the database was left in an inconsistent state... if it understood the constraints well enough. If. I wouldn't want to gamble my production database on the AI's understanding and testing of all the constraints, though.
It's orthogonal.
To be clear, I'm not suggesting blindly deploying an AI-written spike implementation to production, but rather using it to elicit information for better designs.
The fact that a probe that goes off and modifies tables X, Y and Z to achieve the feature gives information for an AI auditor to look for other uses of X, Y and Z, and discover things humans may miss, because with good guidance and a proper harness, AI is usually more persistent and thorough than people. It can turn search results into a checklist and the harness can track completion, and so on. I am far from convinced that your example would not be found via this route.
But if you're doing a spike, no, don't do a design document for it. How can you? You don't know what the design needs to be yet!