Neither of my airplanes is ailing. The Cessna 182 (see my avatar) is powered with a Continental engine which does not use a Bendix starter drive. It uses what is called a Sprag clutch, which does not have the same failure mode as a Bendix. It is extremely unlikely that the starter drive on the Continental engine can fail in a manner where the engine would back-drive the starter.
My other airplane is powered with a Lycoming engine which uses a ring gear on the back of the prop, and a Bendix engagement. However, the original compound-wound segmented-commutator starter motor was replaced with a modern, lightweight starter with permanent magnet field. In the thirteen years I have owned this one, the Bendix failed once, leaving the starter gear engaged to the ring gear, but I knew instantly that I had a problem.
The reason I knew I had a problem is because I had previously wired a LED indicator back to the cockpit. The LED is intended to sense the voltage applied to the starter via the starter relay. Another common failure mode is welded contacts in the starter relay or a bad key switch, where the starter remains powered via the start relay even though the start key switch is released. In the case of the stuck Bendix, the back-driven permanent magnet starter ran as a generator so even though the starter relay broke the connection to the battery, the voltage produced by the starter lit the LED after I release the start switch. I shut down the engine, and found the frozen Bendix.
(See attachment, below)
I participate in several aircraft owners forums. On one of them, one of the posters just had a stuck Bendix on a Lycoming engine but he has the older wound-field type starter. That resulted in a catastrophic failure of the starter, which cost big bucks to repair. He did not have my LED indicator, but the question came up, would my little circuit have indicated his problem. It depends on if there is sufficient residual magnetism in the starter for it to produce a few volts at a few mA to light the LED?
I am going to borrow an older starter from an aircraft mechanic that lives further down the runway. I will spin it up on the workbench using a battery, with my scope connected to the starter terminal. This should show me what happens during the time that it takes the starter to spin down after I disconnect the battery. Only problem is, even if the one I test does it, would they all do it? Would they all produce the same polarity?