Any idea how stock Android handles it?
It's something that's always bugged me. I've never used stock android to compare though.
But yes, a gallery option would be good.
The baby monitor app I use has the share sheet, and I usually have to share it to InShot (or another editing app) to then be able to save it to my gallery.
Even the file manager apps don't offer an option to save it locally, they just give options to upload or share it to a network or local device.