I am just finishing my first year in grad school. I wanted to comment because I seem to have a slightly opposite view than the previous one. First, I should admit that GRE subject problems are not easy (at least for me) and the time constraint makes it harder. I hated the experience of taking GRE's personally because I am not a good test taker to begin with and thought it was pointless. I enjoyed slow math. By that, I mean I much prefer tackling a certain problem (set of problems) I enjoy and thinking about it thematically over long periods of time. Nonetheless, I have managed to get 70% after intense study, and thought that was it just for the sake of applying for grad schools. But now that I am in grad school, I realize that the learning skills I picked up from studying for GRE was an important (if not necessary) asset to have to go through grad school. Working on assignments or preparing for qualifying exams require extensive reading on your own (outside of classrooms) to digest the materials. I think being able to read diligently and absorb new materials (whether it is a topic of your interest or not) is necessary. If reading is an obstacle (as it was for me), it is a challenge you should overcome and it cannot be avoided. (Also by the way, the materials from GRE tests are basic knowledge that will frequently appear throughout grad school).
Now, I don't write this to discourage you. I wanted to give you an insight into what it is like to be in grad school (at least in my experience) so that you are prepared. Of course, what we want eventually is not just memorizing some materials but producing fruitful research, and that's the day I am hoping for (pass my qualifying exams and actively work on my research). That is the reason I stick around and patiently try to overcome my barriers, and I think it is what we all do as phd students.
I am also an international student in US, and I know it makes getting into the grad schools more challenging. In my opinion, there are plenty of great math professors in US, hence plenty of opportunities to receive great training. So,don't feel like you have to get into only the top schools. I think you should focus on finding what schools might be a good fit for you (look up the people in the department, what research they do, etc.). I wish you best luck