Valve has confirmed that the UI from Steam Deck will replace Big Picture mode on Steam.
This was discovered after Valve employee austinp_valve responded to a question in the Steam Big Picture forum about the future of the interface.
Big Picture was released in 2012 as a console-like interface for users who plugged their PCs into large monitors and TVs while using a controller as opposed to keyboard and mouse. It became the basis for SteamOS which was used for the ill-fated Steam Machines.