The best explanation for a slide in a formula car was given to me by a Jim Russell driving instructor, I'm not sure where it originates.
"A open-wheel formula car acts like a dart flying backwards"
What I got out of that is; once it wants to turn around on you, you're probably already backwards.
Corrections need to be made very early and very smooth. The corrections are usually made before you can visually notice the car sliding, the steering wheel movement is very small and quick. The return of the counter-steer is also important. It needs to be quick as well, but it needs to build up to the original steering angle or the rear will snap again. Concerning the throttle and brakes, unless it is power oversteer (which is not a common issue in the FOX), the pedals shouldn't move. They shouldn't move because you will upset the pitch and fore-aft weight transfer, which will make the cars grip less predictable and much more difficult to correct.
Sometimes the correction or "catch" isn't even counter-steer, rather its just less steering angle and a pause on the pedals. This lets the tires catch up with what your asking them to do, with out upsetting the car too much.
It takes lots of time to get fast and comfortable in a open-wheel car. While its difficult in all cars, open-wheel cars are particularly challenging regarding correcting mistakes without causing extreme negative results. Theres nothing you can read to teach it to you, its muscle memory, hand-eye coordination and just plan experience. After a while you will notice your hands moving with absolutely no apparent thought crossing your mind of a "slide."
I'm no expert, but I'm fairly sure all of that is, more or less, some good advice. Keep driving
Concerning the left foot braking idea, well it might work in LFS, but in real life your more or less asking to continue off the track at that angle right then. No more the bias, brakes transfer weight to the front, increasing the grip of the front and reducing the grip in the rear. Locking the fronts takes away nearly all the grip in the front and by doing that stops the slide because of the balance of grip front to rear. Or, it may stop the slide because it locks up all four tires, which stops and reduces all control and grip the tires had, so the car holds that angle and slides wherever its heading. If you ever talk to any driving instructors, the last thing you want to do is abruptly lift and even worse than that, would be brake "jabbing." If you've ever seen a car spin because a car in front spun its because the car behind abruptly lifted and/or slammed on the brakes, which makes the weight go to the front and thus, spins the car right around. Braking to save a slide should only be used in circumstances where the car is no where near the point of recovery and if it wasn't used the car would for sure quickly loop around and go off the track. Thats why at most driving schools, they pound the idea of two feet in. You start to spin, slam on the brakes and if you can the clutch. This makes the car slide in which ever direction its heading very predictably so that people behind you can make a decision on where to go around you. It more or less eliminates the possibility to smoothly catch a slide in my opinion.