@Carl 

OK, I double checked the connection.
The switch is definitely connected to the Line, not the Load, side.

I also deleted, then reinstalled the app.

The firmware of all the devices is up-to-date.

The Kasa app is version 3.1.5.

Now here is the interesting thing.

After I wrote this (and taking some time to calm down a bit), I decided to give the app one more try.

Of course, now it seems to be working.

There was one instance where it needed to be toggled twice, but I suspect that I may not have waited long enough between tries.

So for now, it appears that things are in order.
Thanks again for your feedback.  Hopefully I won't have to revist this topic again.

  @Carl 

Here's an update:

I installed a second 3-way switch for another light (not on the same light as the first switch so only one switch in each circuit).

I ended up having the same issue as the first one.

Deleting the app, then reinstalling it made everything work.

This was fine for about a day.

Today, the last installed 3-way switch works as expected, but the first installed one now works like it did before the "fix".

It is going to be a real pain if I have to delete and reinstalled the app every other day.

Is there any way to make things more stable???