If it's a big enough gap, I usually fill it with small pieces of a similar looking wood and wood glue. I tap them in with a wooden mallet until they're snug then sand it after it cures.
I think you are on to something here. Saw online somewhere that thin veneers can be made by edge planing a board. The strip can then be glued and wrapped around the tenon to take up the extra space. Think this is what I’ll try.