Aircraft carriers do not depend on Washington

Would aircraft carriers still be built if the Washington naval-treaty system never appeared?

Yes. Carrier construction was already underway during the First World War, including Hermes and Hosho.

Carrier construction was already under way during the First World War, before the Washington treaty negotiations. Hermes and Hosho are the named examples of projects begun before the 1920s. Continued carrier building therefore belongs to this counterfactual's starting conditions.