In my experience where schools / sixth form colleges say that further maths has to be done as a 4th a level, it’s not because they consider it to be a not-proper-full a level, and also not because they think that anyone applying for eg Cambridge maths has to have 4 a levels in total.
It’s usually because further maths a level is HARD unless you are really in the top top group at maths, and tends therefore to have a higher dropout rate than other a levels. So they don’t want to risk people taking maths, FM and physics, then dropping FM and finding themselves with only 2 a levels. Certainly in my son’s class, at least 1/3, maybe more, dropped out of FM in the first term.
Two Maths a levels at once, or one after the other, is very intensive and the pace is hard to keep up with for all but the most mathematically able students. I will never forget being excellent at maths all my life, getting an easy top grade at a level, then being completely lost at uni when the pace picked up and the maths moved into further maths territory (science degree not maths). I’d never imagined finding maths hard to follow and it was the combo of pace and content I couldn’t keep up with. Guessing this happens during maths + FM a level combo for some students and so they drop one.
At specialist maths 6th form colleges, they’re more likely to be comfortable with 3 inclusive of FM because the criteria to get in is likely higher.
I also suspect though that the drop out rate at places like my son’s college is a somewhat self fulfilling thing - since all the FM students are taking 4 a levels by definition, they’ve all got a higher workload and are more likely to need to drop something.