As Brandon from Webflow says, it is possible to create a new CMS item via API and push it live.
However, it is not possible to update an existing one and push that update live.
This feature though is crucial when using Webflow for an e-commerce site where regular synchronizing (of stock availability, prices etc.) is required via API. Currently, the only workaround seems to be to force Publish Site via API, which may not always be desired though.
Please, please consider adding this... Thank you! :)