No worries Bryan, I understand your confusion... so let me clarify...
Firstly, the breath. To meditate on the breath is usually performed by putting the focus on it, and for most people they mistakenly think about what it's doing... "iiiiinnnnnn".... "oooouuuuttttt" (or whatever) and this type of focus, ends up with the mind trying to control and often slow the breath down rather than just letting it flow naturally and simply observe it.
Now, the mantra. Yes, the mantra is a tool, and it starts as a construct of the mind, you are correct in that. So, in a sense, it's the same as using the breath, because we can construct it in mind and put all our focus on it.
So, I suppose I'm being somewhat judgemental, but that is from observation of how many do focus on the breath as an object and let their mind become attached to it rather than simply observing, whereas from experience, most who use a mantra (not something most aim for initially so it's often taught correctly) don't get attached to it and just observe it. Of course it varies from person to person.
In truth, either way can be used for meditation, but the key is in the person understanding the difference between attaching focus from the mind to the "object" compares to simply being aware of it.
From personal experience, observing the breath is something the mind is familiar with, and something it can become easily attached to, yet a mantra, especially one that has no meaning to us (so a mantra such as "hiram" rather than "I am One") is more easy to simply observe and be aware of rather than become attached to.
Of course, everybody finds the technique that suits themselves, and there are certainly proficient meditators out there who use breath or other things as their tool of meditation.
Hope that clarifies my understanding to you. If not, I'll try and clarify more later.
Hugs
Giles