If you set up "winit" so it only redraws on events, then you don't trigger an EGUI redraw very often. But if updates that affect the EGUI items are coming in from another source, like the network, you have to explicitly trigger a winit redraw cycle.
Don't know about web. Seems overkill if all you have are GUI widgets. Egui is most useful on top of something more graphical.
Scrolling text is a problem. EGUI isn't good at big text boxes with much offscreen text. That can be worked around.