Success!! I was at this for a while before I asked the question, then within less than a half hour after posting the question I solved it. I found this page: https://www.lg.com/eastafrica/support/video-tutorials/CT20279058-20151100498340-others. From there, it clicked that “apps” are the things at the bottom of the screen that appear across the bottom of the picture when I click the home button on the remote. I clicked on “Amazon Prime,” and I was prompted to update the app. After doing that, I got a screen that guided me through the process of using my computer to pair my TV to the Amazon account. After I did that, the movie I purchased (plus a whole bunch of other things they would like to sell me) popped up on the Amazon Prime home screen. But clicking on the movie I purchased did nothing. I don’t know if things were still loading behind the scenes, but I grew impatient after a few minutes of the movie failing to launch. Nothing at all seemed to happen when I clucked on it.
I went back to the web browser and tried again via that approach, with the same result I had earlier. Eventually I went back to the Amazon Prime app and clicked a few things. Eventually, clicking on the purchased movie worked. Who knows why? I’ll never understand. I’m too old, and it seems that doing the same thing multiple times eventually produces a different result. This is why technology frustrates me, but at least a few minutes ago, it worked for me. This time.