For the past few months, I've been repeating "I" and "am" silently coordinated with successive inhalations throughout the day. This practice has been immensely beneficial with respect to work and daily living. But will this lead to insight?

Can I reach third path by simply repeating a mantra coordinated with the breath? 

howdy,
it sounds like you are using (at least partially) the technique advanced by Yogani at aypsite.org.

its interesting and effective.  i used it several years back.  it is a concentration technique and brings one throught the jhanas to "no thing ness" IMO.  there is definitely potential for insight at this leve of concentration but it is not the focus of that excercise alone.

its a silent mantra technique (on aypsite.org) and it is well known that one can get very concentrated very quickly using mantras, so my take is that while your jhanic abilities will improve, investigation needs to be carried out as a supplement to that to get path.

let us know how it works out.

Actually I first heard of this practice via listening to Robert Adams (an American Advaita teacher in the Ramana tradition), who was likely around before contemporaries used some of his stuff.
I've used "I" "am" as a mantra, but it didn't really 'do it' for me, since it turns into a concentration practice, and IMO, focusing on the "I am" feeling/self/object is more of an insight practice (like self-inquiry) which can lead to anatta insight etc.

But purely as a mantra, I guess it could work for some, as any other object (which might then lead to concentration type results, unless a shift in focus such as the above was detoured into).

Actually, I got it from the teachings of the Ramana Maharishi. He says one must hold on to the 'I' and obliterate thoughts. He also said that pranayama helps to silence thoughts, and that watching the breath is a form of pranayama.

I have been practicing this form of self inquiry and I've noticed that when I hold on to the 'I', the breath becomes finer and smoother. I become relaxed, free of stress, and sitting still is easy. 

great.  to answer the question in your first post about being able to reach 3rd path with your mantry and breath coordination method the answer is theorhetically yes.  there are suttas which state that one can go "all the way" with meditation on in and out breathing alone.  sorry i don't have the citations at hand.

that said, there would have to be deep investigation going on IMO. are you doing investigation or going for deep jhanic states?