If you publish a piece of software under two licenses, one open source and one commercial, why would anybody pick the commercial one?

You cannot say "corporations have to pick the commercial one". The only way to do that would be to include a term in the open source license so it cannot be used by corporations. But then it would no longer meet the open source definition. "Free for private use but payed for commercial use" is not open source!

But if the open source license is copyleft (e.g. GPL), any software that uses it would also have to be open source. So dual licensing allows corporations to pay to keep their own code a secret.

This implies that dual licensing is pointless for non-copyleft licenses like MIT, and also for developer tooling that is not directly used in the final product.