Why are cumulative tests typically weighted so high (U.S.)?
Cumulative tests show that you understand everything so far; they are meant to help students who perform poorly on the first exam, so they can do better on the second one, since they will have an idea of what to expect on the second one. It is uncommon when a student does worse from one test to the next.

Also, going back to projects, some professors don't value homework/projects as much because any student can copy / get help for their projects. Some professors I know would even give 100% for homework/projects just for turning them in, as they are meant for students to practice and learn by themselves. They are meant to help students prepare for tests. Because of this projects and homework are not a good (trusted) indicator of how a student is doing in a class.

By the looks on your grades, most likely you still don't know the topics of your class, if it was a more strict professor / class, just for failing the cumulative final, you would have failed the class as well, as this would be an indicator that you (maybe) cheated on your projects.

Cumulative tests are usually weighted high because in order to do the last topic you need to know the previous topics as well.