In the schools I've attended it was mostly because they didn't want to weigh final exams too heavily (due to test anxiety), and putting too much weight on a fin

In the schools I've attended it was mostly because they didn't want to weigh final exams too heavily (due to test anxiety), and putting too much weight on a final project doesn't scale well when every class you are taking does so - too much crunch time. Over the past few decades most have moved to allowing you to drop your N lowest homework scores which helps a lot with students entering the course with varying preparedness levels.