I agree with UnityParadox and CaptainSlimey. After 6 years with overwhelming response, I think it's clear that people want it. Having to open a screen capture software just to view a replay is a bit excessive. And if you want to replay a certain section to see what went wrong (eg Messed up a pattern on Mania and wanted to learn a pattern) you shouldn't have to record it. Even more so if you wanted to share the replay to a friend.
Weirdly enough, sometimes you can save failed replays in multiplayer. Maybe because the song completes and places a sort of end marker in the osr file. Kinda like a regular replay, but failed. I wouldn't know how osu handles end markers midway through a song without breaking the game.
These are just assumptions on my part, but saved failed attempts are possible as long as the song completes.