A bad Bendix on a starter motor presents itself in a greatly more dramatic fashion than that. How did the starter look when you pulled it? I'll bet it was fine.

Loping idle while cold is generally related to the vacuum logic not adding up. This means a cracked vacuum line or a plugged vacuum valve or something like that. Basically the ECM is getting the wrong manifold vacuum values so it's got the wrong advance but it can't self-compensate by dropping the RPM so it throws it into that regime and waits for something to change. Usually the ECM ignores that value once temperature has risen to operating values so the problem goes away once it's warmed up.

It's become increasingly common for manufacturers to lower weights by going with smaller batteries and smaller starter motors, which requires some sort of intake/exhaust bypass to lower compression pressure. This bypass is often centrifugal or solenoidal and closes once the engine hits idle. If it doesn't hit that idle quickly it may hammer, which often sounds a lot like your problem.

The hose that's compromised is small, non-obvious and deep in the engine. Should you find it you will never be sure if you fixed it, and you may fuck up two or three others in the process of diagnosis. This is a great thing to farm out.