The chain always falls off toward the bike frame, off of the smaller front cog.

It's likely not the rear derailleur since I'm usually in the middle rear cog.

It might be the front derailleur since when I put it as close to the chain as possible, the chain doesn't fall. This is again, on the third cog up in the rear and the smaller cog in front.

That said, doing the above might just be preventing the chain from falling because of its potential real problem: chain, crank set, or cassette wear.

What are your opinions?

My opinion is you should read this-

https://www.parktool.com/blog/repair-...ur-adjustments

Check and make sure all of your cog bolts are tight, and that your small cog is not bent in some way, then, if that's not the case, check your FD limit screw, and position of the FD to the cogs, angle and height. This will get you started.

Try this:
https://www.probikekit.com/bicycle-ch...FVQV7AodyiUABA

You should of course make sure that your derailleur is set up properly as advised above, but if would like to install a chain catcher anyway (I always do) I recommend the N-Gear Jump Stop®, if it will fit your frame.

The first step is to determine if the FD itself is causing the chain to fall off. If the outer plate of the FD is touching the chain when on the granny and inner half of the cassette, then you have an FD adjustment issue, most likely the inner limit screw. It could also be a trim issue (cable length adjustment) if the shifting and trim on other chainrings isn't spot on.

If the FD isn't touching the chain, and the only way to prevent it from coming off to the inside is to have the FD keep it on with the inner cage plate, then you have a sprocket problem, most likely a single bend or dinged tooth.

Either way the first step is to determine why the chain is coming off, and pinning it either on the FD or the sprocket engagement. It's also possible that a damaged link in the chain itself is part of the problem, but this is comparatively rare, and my money is on the FD trim.