IS helps with autofocus for moving subjects, too; whether in single shot or Servo, it keeps the image projected on the AF sensor more stable and removes extra workload from the AF algorithms while focusing on a subject. The camera is doing phase comparisons as fast as it can to determine the focus plane, and if the subject moves significantly between samples, it has to ensure it's on the right AF point (which may require a switch, depending on your case settings) before sampling again. Holding the image more steady helps the algorithms be more efficient.
That said, for my use (sports/events) I find the wider aperture more important because it improves the accuracy of the AF on cameras like the 7D mk II and 5D mk III (and 1D series of course) with dual cross-type points. Especially in low light situations, the improved accuracy can trump the benefits of IS. You have to decide that. The price of the f/4 IS has come down enough now that it's a reasonable alternative to the mk I if you choose it.
I have found the 24-70 f/2.8 mk I entirely sharp enough for every camera I've used it on, and well worth the savings of $1000 vs. the mk II. There's enough sample variation in both lenses that a good mk I may well be sharper in the center than a poor mk II. I have never seen anything in my photos that makes me want to upgrade yet.