What I discovered was that in order to change the keyboard I had to press my finger into an input text box and hold it down long enough for an "Input Method" dialog to appear beneath the input box as shown above.
You'd think that you could do that in any text box, such as the text box for inputting URLs in the browser. And you'd be wrong.
Well, it turns out I'm partially wrong. As you can see in the handset capture to the left, I'm able to select the input method in the browser's navigation bar. When I went looking for this capability the first time, I couldn't make it work no matter what I did, which is why I had to exit back to the Google search bar on the handset's top-level panel.
And from what limited testing I'm able to do you can perform this input method selection from just about any input text box. But that still doesn't excuse the fact it's only available via a counter-intuitive activity, one which I find I'm increasingly activating by accident (well, it's either that or the "Share page/Copy page url" dialog") when I keep my finger down too long. I know I'm getting old and slow, but I'm not that old and slow, at least not yet.
Android really seems to go out of its way to incite a strong love/hate relationship with the end user, as in I'm really loving to hate on Android. Apps I can forgive, but not core UI problems. Oh well. Just more motivate for me to double down and learn Android internals.