All good advice, except to say, your fitter should have explained the dynamic here; you are not supposed to be able to touch the ground at a stop, until you lean your bike and put a toe down. This applies even if you are using flat pedals. If your seat is low enough that you can put your feet down flat, well, then your seat is too low! I know that sometimes people keep the saddle low at first, as they get used to the movement of mounting/dismounting, but then it is raised very gradually back to where it should be, otherwise your knees will pay the price. The motion is so ingrained, once you get it, you don't over think it and it happens "all at once."
I have the "campus" pedals on my second bike... one side spd cleats, one side flat. They also require the motion to flip into the right position and it's a pain. I rarely ride without bike shoes and believe me, these pedals are going when I trade this bike in for something else. There are lots of dual sided pedals. I started with dual entry spd pedals on my first road bike, and then went right to Speedplays on the second road bike, which also are dual entry.