The fact that intro math classes don't do proofs (Type B) is because of the same pressure from people who only want to do Type A.
Due to internal changes in my uni, for the first time, my freshmen year, the math department taught proper proof-based Calculus 101 (from Apostle of all books) to all majors. Then the engineers and biologists complained so much, they had to cut out a lot of proofs from Calculus 102. There were even more complaints, so by second year, there were hardly any proofs in the core math courses. In a few years, the calculus courses had become devoid of proofs.
Some unis have separate intro courses for math majors, but it's very difficult to offer them in the current economic climate.
I think Proof vs non-proof is part of it, but it’s mostly related to level of abstraction.
You can do proofs for calculus, probability, or logic and still feel like you’re working with the types of problems you do in type A math.
But once you start doing proofs in modern algebra or topology you’re doing things with abstract objects that seem to exist for the amusement of mathematicians that look down on “applied math”
Due to internal changes in my uni, for the first time, my freshmen year, the math department taught proper proof-based Calculus 101 (from Apostle of all books) to all majors. Then the engineers and biologists complained so much, they had to cut out a lot of proofs from Calculus 102. There were even more complaints, so by second year, there were hardly any proofs in the core math courses. In a few years, the calculus courses had become devoid of proofs.
Some unis have separate intro courses for math majors, but it's very difficult to offer them in the current economic climate.