The Canon 70-200 f2.8 (NON-IS) is a parfocal lens; it maintains focus while zooming and thus provides an excellent benefit for shooting video with live zooms.
I had EF 28-135 IS which was not L but could be something to consider - acceptable picture quality, IS, very good mm range and affordable price. I am sure you would achieve better results with it in video than 70-200 f4 no IS.