Thanks to a contribution WP Browser can now “drive” real browsers.
Solving my own problems I was happy enough with WP Browser extending Codeception own PhpBrowser to do its job. But that’s not the case for everyone and the lack of any JS-capable browser testing did limit its possibilities.
GitHub user maxchirkov contributed to the project creating a new tool called “WPWebDriver” that allows using the same methods available to WPBrowser to be used over an extension of Codeception WebDriver module.
I will be playing with it in the near future and will post any experience/findings.