If the system isn’t built to move it from memory to the disk then it isn’t easy to just move it over. The browser might have the file stored in a way that makes it easy to stop and fastforward or even allow it to play or partially load,. Trying to save a file that might be incomplete is far more complicated than a good old reliable direct download.

Think of it like orange juice. The browser may have stored it in a wide shallow bowl because that works better for the browser, but it would be a pain in the ass to pour into a small cup. Much easier to just pour a new cup of orange juice when it is in an expected format.