Thanks, all, for the suggestions. The latest: I removed the Radeon GPU, uninstalled all of the Catalyst software, and restarted.  No change. I moved the trackman mouse to another PC, and it worked fine there.  Ultimately, this PC will end up with a Logitech wireless keyboard/mouse combo I bought at Costco, so I plugged in the bluetooth dongle, and all seems to be working correctly now.  I feared that adding Bluetooth to the mix would just add one more thing to potentially go wrong, but so far, so good.  Seems to have "fixed itself" by swapping the mouse, even moving the mouse to another PC hints that the mosue itself is fine.  Go figure. Next step was to put the Radeon back in (for hardware acceleration on video decoding).  Still all working fine. I still don't know if it was the mouse driver, the mouse itself, the port, some combination thereof, or something else, but it's OK now.
I have an XPS 8700 (c. 2013?) I'm repurposing. I wiped the drive (Samsung SSD) clean and re-installed Windows 10 (it's currently running 21H2). The only extra hardware is the Radeon 7570 that came in it, and an Asus Xonar audio card (I needed optical). I'm using the onboard graphics to run a Samsung ultrawide monitor (5120x1440), and the Radeon is driving a Dell 2005FPW. The biggest "can't get it to work" issue was that the monitor had to be told to use an older version of the DisplayPort interface in order to recognize the PC, but it IS running now at full resolution (being able to do that was something of a surprise! I thought I'd need a GPU to drive that many pixels, but the Intel graphics is doing just fine). The machine will be used for lightweight productivity, zero gaming. Everything is working great, EXCEPT the mouse, which I'm confident isn't actually a problem with the mouse itself. I'm using a wired Logitech MouseMan Wheel mouse, which seems to work just fine in another machine, so I can't believe it's the mouse itself. What's happening is that when I drag something (a window by its title bar to move it, or by one of its borders to resize it), I press the left mouse button, start the drag, and while the mouse button is still down, and the mouse is moving, the drag operation stops and the cursor changes back to the regular mouse button.  It's AS IF I had lifted the mouse button, but I definitely have not. I suspect some other driver is interfering WITH the mouse, resetting it to a "button-up" state. Device Manager doesn't show any malfunctioning devices. The only thing that's even borderline seemingly amiss is a "legacy device" that appears, but not with a yellow triangle or a red X, so it seems to be "working" properly. Any suggestions?
XPS 8700, mouse driver problem?