Really why? Once your primary fails, you immediately switch second stages. Once that 5-15 second task is complete what exactly is impeding your progress toward the surface?
Seriously how much do you have to think or coordinate or communicate? Are we to assume that we have multiple, simultaneous failures - or problems like entanglement? That seems to be outside the scope of reasonable precautions for a recreational dive.
Are we to assume that our buddy has not noticed, not offered us air, but we need to spend time tracking them down, showing them a problem and making sure they understand? And what benefit would that provide? A courtesy to in an inattentive buddy?
The whole idea of 2-3 minutes on the bottom after simply switching second stages seems "Wildly Pessimistic". Of course if you do have to make your way horizontally to an ascent line or mooring or a safe place to ascend, then that should be explicitly included in the calculations of emergency supply, but in most situations, I am going straight up - there is really no reason to hang around or think about options - I really have one - go up!.