If I install Picroft from an image, none of the other software that normally comes with Raspian gets installed, right? So I wouldn’t have the chromium web browser?
Does chromium come preinstalled? In any case, yes, you’d probably need to install it and a display manager.
On Picroft there is no desktop environment by default. So to get what you’re after, you would probably be better off starting with a complete Raspbian OS or Ubuntu image and then installing Mycroft.
The Mycroft-GUI (as used in the Mark II) can display web pages itself. So depends on what your intended use of the browser is.