I think it's actually cleaner to prove the theorem non-inductively (though I appreciate that what GS asked for was specifically a cleaned-up inductive proof). E.g.: "Count pairs (vertex,edge) where the edge is incident on the vertex. The number of such pairs for a given vertex equals its degree, so the sum equals the sum of the degrees. The number of such pairs for a given edge equals 2, so the sum equals twice the number of edges."

(More visually: draw the graph. Now erase all of each edge apart from a little bit at each end. The resulting picture is a collection of stars, one per vertex. How many points have the stars in total?)