Answer
Does screenshotting a photo remove its metadata?
Yes, for the original metadata — a screenshot is a new file that never contained it. But you pay for that in resolution, and the screenshot carries its own metadata instead.
Updated
A screenshot is not a copy of the photo. It is a fresh capture of whatever pixels your display was showing, written out as a new image. Nothing from the original file — no GPS block, no camera model, no timestamp, no serial number — has any route into it. In that narrow sense the trick genuinely works, which is why it keeps getting recommended.
The costs are real, though. You get the resolution of your screen rather than the resolution of the photo, so a 12-megapixel image can come out at a fraction of its original size, cropped to whatever was visible and re-compressed by the screenshot pipeline. If the picture matters, that is a steep price for a metadata problem with a lossless solution.
And a screenshot is not metadata-free. It is a new file with new metadata: the capture timestamp, the device or software that produced it, the display colour profile, and on some systems the device model. That reveals less than the original did, but "less" is not "none".
The thing screenshots genuinely do not fix is anything visible in the frame. A cropped-in screenshot of a photo still shows the photo. If your concern is a recognisable street sign or a reflection, no metadata operation of any kind addresses that.
If the goal is to keep the picture and lose the metadata, stripping the file directly does that without touching a pixel. How that works is a matter of rewriting the container and copying the compressed image data through byte for byte.
More on this: Metadata remover, What is EXIF metadata?.
Terms used here: Lossless.