Question: I am a Master's student in math who is doing pretty well so far. 
I aced my first two classes (real analysis) 
and my summer session in probability theory is going well, too. 
Even before my Master's study, I already thought of a PhD. 
Nonetheless, there are concerns:

1. My good grades were the result of grit, not brain, 
meaning that I had to spend time to understand what the book is telling me. 
Furthermore, a lot of times during exam and homework, 
I had to try and fail a few times before arriving at the solution.
2. I did not study math during undergrad. Before my Master's study,
I gulped down three semesters of calculus, plus LA and DE in one year. 
So compared to other students, I am already behind. 
In the next two months, I will learn complex numbers on my own
and review my linear algebra.
3. I am a professional in a field that has nothing to do with math or research. 
After the holiday, I will speak to my advisor about taking a thesis class. 
It's not an insurmountable obstacle, 
but, in an unrelated field, I do have less time to concentrate on math.

At the end, how big is the jump from Master's to Phd? 
Especially after what I said in (1), I do worry that it is beyond my ability. 
I do not hold anyone responsible for my decision. 
So, please kindly offer your best assessment of my situation.

Answer: 
As Austin Mohr, pointed out, it is unlikely a stranger on the net fully answer this for you, but what you are experiencing is similar to what I experienced, so I can tell you how it was from my perspective.

Given what JeffE commented, studying through true 'grit', does involve 'brains' as you are developing and applying problem solving skills and stategies - which has clearly been effective, as you mention, you have good grades. So, it seems you have developed effective study techniques.

Point 2 is pretty much not an issue, based on the skills and strategies you developed in point 1.

If you do an unrelated thesis, as you mentioned in point 3 - look at this way, you'll be developing the research and synthesis skills needed for a PhD.

What I found about the transition between the Masters and PhD, was that it wasn't so much of a leap, but a case of using the skills developed a lot more thoroughly.

I hope this helps.
In your shoes, if I were applying for a PhD program, I might be thinking in terms of an "ABD" (All but dissertation.)

From the sound of it, "grit not brain" will get you through your PhD courses. You likely will pass the "comprehensive" examination. And then the fear is that you will "freeze up" when it comes time to write the dissertation because you are behindhand in "natural" (as opposed to synthetic), talent compared to others.

You will have to balance these real concerns against the benefits of "taking courses" and getting as far as you can, before possibly running up against a brick wall with your thesis.

This was basically my story (some decades ago). I was "counseled out" of a PhD program by a dean who observed that I had the preparation and the brains for the program, but lacked the "spark," "thirst," or drive that would see others through, but maybe not me. 

What's worse, as a history undergraduate with mostly As, I had trouble finding a senior thesis topic, and spent the first part of my senior year wondering if I would be able to graduate, and thinking about writing a "trivial" paper that would earn a C. In the middle of the first semester, I stumbled on a viable topic that earned me departmental honors.

I can see one of three outcomes for you. 1) A "light bulb" will go off in your head at some point before you approach the end of the PhD program. If this happens, you're fine. 2) You will somehow squeak through, not quite knowing how you did it, and come out of the process a bit shell-shocked. 3) Neither of the above will happen, as you feared, in which case you might "bail out" as an "ABD."

I can't predict the outcome for you, but am sharing my experience. Just understand what the risks are, take into account your circumstances, and (hopefully) make the best decision for you.