The Westside system is an advanced system, not an intermediate system. As all Westside lifters have said, it's not a static program, it's a collection of principles and methods that are tailored to individuals training at Louie's gym under his guidance.
Westside training seems to be a little less popular now as geared powerlifting is on the decline and the rise of raw/unequipped powerlifting has become more popular. The Westside system was really built for guys who competed in very heavy gear, although there are successful raw lifters who also employ the methodology. I trained at a gym for 3-4 years that was primarily a strongman gym with lots of incredibly strong lifters, everyone of which trained "westside" but tailored to strongman rather than powerlifting. No one trained in gear and they all got strong. There are two gyms here in Houston with some very strong powerlifters that pretty much do straight up westside and they produce some really strong lifters. The coaches are smart guys who know how to manipulate the system to their lifters. So, under the right circumstances the programming works very well.
There are actually two difficult days for each lift, a max effort day for the bench and a speed/volume day for the bench and both use a high volume of exercises as assistance following the main lift. Same for Squat/Deads....one day for speed/volume where both lifts are trained and one day for max effort, usually alternating back and forth between squat/dead variations
Box Squats mimic the effect of the squat suit and yes, they are a little easier to recover from at high volumes in my opinion. There is a lot of GPP work (sled dragging, extra workouts, high volume assistance etc) because they train with some pretty high volumes on Squat/Deads at a very fast pace and you have to be in shape to complete the workouts. Louie also believes that after a certain point that the lifts are built and improved upon by shoring up weak points with special exercises (i.e. assistance work) rather than just training the competitive lifts directly all the time. They do do a lot of regular deadlifts but mainly on their speed/volume day for fast singles. The max effort day is usually rack pulls, deficits, goodmornings, or a squat variation.