"So I worked for a few years on second shift during college at a factory that made architectural doors - mainly veneered with expensive wood species and some door pairs book matched, etc.  Anyway, there are time when the veneer doesn't adhere in the middle of the door face for example and the only way to fix it is to get some glue in there.  Our solution was to fashion a "glue knife" out of a piece of metal pallet banding.  We'd afix a handle out of some beautify scrap wood and then point the banding and thin it out on an electric sander.  Then we'd do just as you said - working the glue in from the outer edge.  Last trick was to have an old laboratory hot plate with various pieces of 1/4 inch steel always hot so we could weight and dry the patch quickly.  We used casein (hide) glue and lighten the dark brown color with titanium dioxide if we were patching woods like birch or maple."