The reason you want to donate a known working reg is the OOA diver may be close to panic, or at least starved for air. You have been breathing from your primary. When you hand it off, you will have taken a breath and will likely be calmer than the other diver, giving you time to grab your backup without freaking out.
I think you can donate a reg a lot faster if it is from your mouth. I dive SM, so I know the reg on my short hose works. Non-issue. If I have to donate my clipped off long hose fast, I just use the breakaway instead of unclipping it.