Well, I got some suggestions from tech support, but not sure they made a lot of difference. However, at the moment it's working correctly.
It doesn't seem to be an issue with signal strength, since the remove worked fine before, and fresh batteries didn't make a difference. Following the normal steps to re-pair the stick and remote didn't achieve anything. But for a while this worked: I'd remove one battery from the remote and put it back. For a short period I could use the remote—usually long enough to get to whatever channel I wanted and choose a program. If I set down the remote for any length of time, even a minute or two, it'd be unpaired again. It was a pain to have to open the battery compartment and remove a battery every time I had to pause or stop play, but at least it worked.
Recently the advice had to do with resetting my router to the preferred channels—I believe I just had to set the 5GHz channel to 161. But again, this was all about signal strength, so it achieved nothing, except that I had to spend a couple of hours trying to access my router's channels and working with Frontier tech support to do it. Since it didn't solve the problem, Roku tech support was going to send me a new remote—but since it happened with both the old and new remotes, it didn't seem likely to me that it was a faulty remote either.
I was about to tell them that, but figured I needed to test the old remote before I did—and in order to do that I had to unplug the streaming stick and re-pair it to the old remote. And oddly enough, that seemed to work! I've used it two or three times since then, and haven't had any issues with the remote not working the first time, or after sitting for a few minutes—or a whole program. So maybe there's a fault with the pairing software that makes some units fail to accept pairing repeatedly—but just some of the time.
If it happens during a program, just removing a battery and putting it back in may jog the remote—but if it's not working at the beginning, it looks like you may just have to keep going through the pairing process until it takes—i.e. remove the batteries, unplug the streaming stick, wait a few seconds, plug the streaming stick back in, put the batteries back in the remote, press and hold the pairing button, and wait for the light in the battery compartment to flash brightly and the remote to pair with the streaming stick. If it doesn't pair after a minute or two, you might need to try again. I just hope you don't have to go through what I did the first time, with the TV refusing to recognize that the streaming stick was plugged in after being unplugged. Good luck!