Have you tried it?
What you're describing sounds like it would work fine to me. The blog post is misleading imho - it implies that SQLite doesn't handle concurrency at all. In reality, you can perform a bunch of writes in parallel and SQLite will handle running them one after the other internally. This works across applications and processes, you just need to use SQLite to interact with the database. The blog post is also misleading when it implies that the application has to manage access to the database file in some way.
Yes, it's correct that only one of those writes will execute at a time but it's not like you have to account for that in your code, especially in a batch-style process like you're describing. In your Python code, you'll just update a row and it will look like that happens concurrently with other updates.
I'll bet that your call to ChatGPT will take far longer than updating the row, even accounting for time when the write is waiting for its turn in SQLite.
Use WAL-mode for the best performance (and to reduce SQLITE_BUSY errors).