I think it is good you are keeping the existing starter. Avoids the rebuild quality problem. But what is causing your problem then? … hmmm … well, the first thing I can think of is that it is pretty unusual for the brushes to go bad on a starter. This is what the auto-electric shop told me anyway. The place where they fixed my Corolla starter. Usually something else fails first. Are you sure the brushes were bad?
Next idea: Does this starter have the selenoid contacts as part of the starter ass’y? Most modern starters are configured like this. If so, these contacts might be somewhat burned or corroded. The increase in resistance might cause the starter gear to not extend fully, so it fails to engage properly with the flywheel. Consider to replace those contacts with new ones. They are usually inexpensive, but auto-parts stores often don’t carry them, and they may have to be ordered by mail. But if you secured the brushes, the place that sold you the brushed may have the contacts too.
Another idea: When you turn the key to “Start”, 10-15 amps of current is supposed to flow through to thin wire on the starter, which powers up the starter’s pull-in coil, and causes the selonoid contacts to engage, as well as pushing out the gear to meet the flywheel. At least that is how it works on my Corolla. If there is extra resistance in this circuit – anywhere between the ignition key switch and the thin wire where it connects to the starter motor – this could prevent this sequence from working correctly. With 10 Amps, even a 0.2 ohm resistance would cause a 2 volt drop, and could make this operation fail or only partially extend the gear. One way to testif this is the cause is to measure the voltage on the thin terminal during attempted cranking. It shouldn’t be much below the battery voltage (during cranking). ( If either the battery voltage or the thin wire voltage is below 10 volts during attempted cranking, that’s usually an indication of an electrical problem before the starter, not the starter failing.)
I had a problem w/my Corolla a while ago where both the ignition switch and the clutch safety switch were failing because of burned contacts and corrosion and just too many uses, which was causing the starter motor to click but not engage. But sometimes it would engage but only partially, and I’d hear unusual metalic sounds like the gears weren’t properly engaging. Replacing both the ignition switch and the clutch switch fixed the problem.