You are free to run the program how ever you would like, and I think you will be able to gain strength in doing so, but I think you'd give up a lot of the benefits of this program.
My impression is that S&S was supposed to be "Strength first", but also be rounded enough to qualify as GPP. In the first phase of S&S, when you are working on building strength & proficiency in the exercises, Pavel suggests that you do some "Jolt" activity a couple times a month. I'm assuming this is to touch on your glycolytic & aerobic systems using other activities. No timed test to avoid adding speed to dysfunction. Pavel assumes by timeless simple that you've reached a level of proficiency where you can push the density with timed tests to start getting into the other energy systems. I think the demonization of glycolytic training is unfounded. You need to do it, just not at the intensity & frequency to derail strength training. When done in the right proportions, glycolytic & aerobic train can synergistically help your strength.
I think Pavel's recommendations in 2.0 are very generous as far as limiting glycolytic training. When I was doing S&S, pre-2.0, I would use a timed test session with a lighter bell as my "light" day. It was my favorite training day. It was over quick, and it gave me an energy boost the rest of the day. I would usually have pretty productive training on the following day. It would be to your benefit to start these weekly glyco sessions as soon as possible (after you hit timeless simple). It's not gonna get any easier when the bells are heavier, I'd start now if you can.
Good Luck!