You are correct. I pointed that out in one of the other posts in this thread. I don't blame you for not reading them all.
The use of the formula relies on the user understanding that the correct sign must be used for GRADE.
The formula only deals with the friction, f, at the tire/road interface. The formula can be used to solve for any of it's variables based on knowing the other variables and plugging them in to get a solution. It also needs to be kept in mind that the way it's often used, and how its being used here, one simply makes an assumption that is said to be the f for the road. That's a bit simplistic because there is no "f' for the road. f only has meaning relative to what ever material is the other half of the interface. In this case it's a tire. And different tires can produce different f's, sometimes significantly different. Also, as you mention, there is the issue of whether the brakes are locked. If they are locked and the tires skidding you have a different f value then what you would get/use if the tires were not skidding. F also varies with the speed of the things moving against each other at the interface, so the f value is an average of all the f's as the speed goes from the starting speed down to zero.
Or are we