Which is the more competitive path to grad school for mathematics?
Let me start by saying that while I have never been involved in graduate admissions (my college does not offer a graduate program in mathematics), I'm an assistant professor of mathematics at a well regarded liberal arts college whose graduates regularly go on to attend top graduate programs. 

As you acknowledge in your question, the reality of the situation isn't nearly as black and white as your dichotomy seems to make it out to be. In my experience, the most important aspects of your graduate school applications will be (in no particular order):


The classes you took and your grades in them. Since you say your goal is to attend a top PhD program, you should be taking challenging courses and doing well in them. But of course you knew this already. There is a certain balancing act that one has to pull off in selecting what courses to take. Essentially, this boils down to breadth versus depth. On the one hand, you can take lots and lots of fun elective courses in all sorts of cool areas of math. On the other hand, you can choose a smaller number of areas and go really in depth with them, taking the introductory graduate sequences in them if possible. When I was an undergraduate I took the former route and took 24 math courses (I just counted) during my four years of college. These courses (some graduate, some undergraduate) were on topics like mathematical logic, algebraic geometry, primality testing, etc. I don't advise my students to do this, but I'm glad that I did and to this day find it useful in my research to know little bits about a wide variety of fields. Assuming you are interested in pure math, I'd advise you to go the second route and focus on algebra, analysis and topology. You'll likely have to pass qualifying exams in these areas in grad school before you can advance to candidacy, start working on research with an advisor, etc., so the stronger your undergraduate background in them the better. Your application is going to have to convince PhD admission committees that you'll be able to succeed in their programs, and having already done well in graduate courses in core areas (i.e., algebra, analysis and topology) is going to make this much easier. 
Your letters of recommendation. It goes without saying that to get into a top PhD program you'll need strong letters of recommendation from professors that know you well. Ideally these will be from people that have taught you in multiple courses (ideally advanced courses or independent reading courses), mentored you in an honors project, or mentored you on a research project. Something that is crucial is that the professor know you well enough to be able to write a strong, detailed letter that says more than just "This student got an A in my class." I find it extremely frustrating when a strong student asks me for a letter of recommendation for graduate school when I've only had them in one course and they never really attended my office hours regularly or do anything else that in that course that let me get to know them at all. In cases like this it's hard for me to write anything but a generic letter that might not do all that much good. Something else to keep in mind as you try to game the graduate admissions system is that good letters are going to compare you to other undergraduates that the letter writer has interacted with. As an example, when I write a letter for a student applying to top programs I'll say something like "Of the roughly X students that I have interacted with during my time at Oberlin, this student is most similar to Students A and B, who are now graduate students at Stanford and the University of Michigan," or "This student is one of the 3 or 4 strongest in their (very strong) class." This kind of thing lets the letter writer make it clear whether you are a very strong student, an exceptionally strong student that is at the very top of their class, or the type of student that only comes along once every few years. If you are going to be graduating from a university that regularly sends students to top programs, having letter writers that can make these sort of comparisons could be especially important.
Your score on the GRE (math subject exam). I don't have much to say about this one. The fact that programs receive many more applications than they have spots to fill in their incoming classes means that scores on exams like this are a very convenient way of comparing people and/or developing cut-off scores. That isn't to say that you won't be able to attend grad school if you don't do extremely well on the exam. It is, after all, a single exam, and one that is primarily (exclusively?) taken by people wishing to go to graduate school in math or a related discipline. I've found that students scoring around the 60th percentile with otherwise strong applications haven't had problems being admitted to good graduate programs. If you're coming from a well known university with letters of recommendation from letter writers whom graduate admissions committees will have heard of, then places might be a bit more forgiving than if you have a low GRE score and are coming from a lesser known place with letters from people that admissions committee members will be unlikely to have heard of. 


Returning to your two options, I don't think either of them is optimal. If your goal is to attend a top PhD program then I think that you'll want to find some sort of a compromise between these options. Here's one thing to consider about your Option 1 though. Let's say that you get stellar grades in all of the required math courses at your university, and perhaps take an extra class or two, but nothing else. I'm far from convinced that this will lead to the types of letters of recommendations that will get you into a top PhD program. The issue is simply that I find it unlikely that doing this will lead many of your letter writers to write things about your being at the very top of your department's graduating class, doing well in the most challenging courses your department offers, etc. At the end of the day many of the students that will be admitted to top PhD programs will have done very well in a lot of very challenging courses. Taking a spot from a student that has done this when you've decided that you're only going to do one of the two (get really good grades or take really challenging courses) doesn't seem all that likely.