What's wrong with the good old Bendix gear?
The Bendix gear works well on Lycoming engines because the ring gear is on the outside of the engine. In truth, even for an automotive engine, the ring gear and Bendix are not actually in the engine, but rather in the bell housing between the engine and transmission, where failure of the gears cannot create trauma inside the engine.
Continental and Lycoming designs differ in philosophy in two major ways: Camshaft location (down rather than up) and starter arrangement (internal drive, rather than external). You would not want the huge engine case which would be required to house the ring gear inside at the back of the engine. The Continental engines were favoured for their comparatively sleek front case (with no ring gear) so cowls could be similarly sleek (like those of the C 310). The cost was the need for an internally driven rear case starter.
Continental did use a quasi bendix internal arrangement on the C-85, C-90, O-200, and O-300, with the "pull clutch". Generally, if set up correctly, these work very well, and last a long time with no maintenance. However, if they are set up wrong, they will clash gear teeth, and fill the engine with bits of hardened steel. But this pull clutch lacked the capacity for torque for the biggest engines.
The starter drive adapter is a cunning design, which has the desirable characteristic that in the case of a kick back during start, it may slip, or break the clutch spring, to save damaging something else in the engine which is much more costly. The starter adapter design has only two flaws: It is intolerant of poor maintenance and poor operating technique. The maintenance is well defined and understood by those people who should be doing it. The problem is on the operation side, the planes are not being presented for starter adapter maintenance when they need it.
So operation: Pilots of "key start" Continental powered aircraft. These include the later C-150s, C-180's and larger, and other non Cessna types. If, when you start the engine, there is slipping of the starter (you can hear it running, but the propeller is not turning crisply with it), abandon the start, and have the starter drive inspected and repaired as needed. Do not run the engine any more. Continental and Cessna probably did not think that they needed to say that an aircraft with an engine which cannot be started normally, is not airworthy - so don't fly it! They did not point out this rather self apparent fact, so I will.
Other than for a battery which is simply discharged, do not hand prop Continental powered aircraft when the electric starter will not work. The plane is not airworthy, don't fly it - sorry.
All key start Continentals rely on friction for the starter to work properly. This friction is best obtained within the intended design: A warm or preheated engine, so the oil is not like grease, and an adapter which conforms to the design (not worn out) so the friction is not being obtained with silly amounts of heat created by a slipping clutch spring - by a pilot slipping the clutch during a start until it finally overheats and grabs to start. So those are operational things that the pilot can do to greatly prolong the life of the starter adapter. Failure to do so will be very costly, and probably inconvenient (starters don't fail at home base, if they can fail when you're away!). If you the pilot have the slightest reason to think that after the engine started normally, the starter motor kept running (electrical anomaly) shut down, and ask for maintenance. The other thing to know as the pilot is that it is easy to overheat the starter motor by too much cranking, and this will potentially ruin the starter motor, and risk introducing the stiffness, which will then destroy the starter adapter inner workings. Though it is rarely spelled out, the C 310 I used to fly did - and it's about the same for all starter motors:
Crank for 30 seconds, rest for a minute, crank for 30 more seconds, rest for 5 minutes, crank for 30 more seconds, rest for 30 minutes. All of this is to allow the starter motor to cool.
On the maintenance side, the only really important thing for the maintainer to know that is not published in Continental maintenance instructions, is that overhauling the starter adapter, without also overhauling the starter motor is trouble. Installing some "geared" starter motors (which are Grrrr, PMA approved) can similarly create problems for the adapter. The starter motor must have the mechanical freedom to unwind a fraction of a turn after the engine starts. This allows the starter adapter clutch spring to unwind a little and release from the friction drum on the starter shaft. For newer design starter adapters, this is not so vital, but still a good idea. So a old gummed up stiff starter motor, or a gear reduced one which is stiff will ruin the older style starter adapters (the ones which you can turn the prop backwards with no resistance). The informal way of knowing how any starter motor will work with the older style starter adapters, is to turn the drive tang on the starter motor by thumb and finger - if you can turn it this way, you're probably okay. If too stiff, due to age/condition, or being geared inside, don't use it on the engine.
All that said, a well maintained system is entirely reliable, and a fine design. At least you can hand prop one if you run your battery flat (which I have done several times on carburetted 470s and 520s). If you run the battery flat trying to start a Lycoming, a hand prop start is probably impossible, as the Bendix remained engaged, and now prevents a hand start at all.
That's your Continental starter lesson for the day. For more information, I recommend the linked information from Canadian Aero Manufacturing. When it comes to the starter drives in the C-85 thorough O-300, they are similar but different. I won't drift this thread, but know that they require maintenance and proper operation as well.....