I company I worked at had 20 or so sites on a headless API based CMS. Basically you’d enter structured data in their proprietary backend, and the site would request that data as JSON via the API. The marketing department hated it with a passion, because there was no way to do “one-off” things without involving IT, and IT hated it because marketing needed permanent handholding for the simplest of things.
After about a year of paying €8000 a month for hosting they replaced it with one really good 100% custom WordPress theme.
WordPress has a publish button, post scheduling and revision history all out of the box with no plugins. That right there eliminates most of the need of IT in day to day publishing.
Headless CMS may have that feature but then you have to deal with a more complicated deployment pipeline that a marketer might not be as comfortable with as WP.
There might be a misunderstanding here. Headless CMSes don't mandate deploys, that's only if they're used in conjunction with static-site generators. A headless CMS is just a CMS that provides content as structured data over an API rather then pre-bound to a rendering layer. You can use WP as a headless CMS if you use its content API.
After about a year of paying €8000 a month for hosting they replaced it with one really good 100% custom WordPress theme.