That final part isn't too difficult... the glue is already "dry" by the time you put the spacers on it. The messy part is applying to contact cement so it doesn't get all over the edges and/or edge banding while working on the big surfaces. Back in the 60s and 70s it was typically applied with a brush and a squeegee type implement (often a scrap of Formica). The squeegee was to spread it on the big surfaces quick while the brush was to apply the cement up to the edges of the surface.
The overwhelming majority of users would brush parallel to the edge like you would cut in paint. That almost always resulted in the glue spilling over the edge, or being brushed over the edge. Once over the edge it was likely to gum up the router bits, especially the bearings on the bits. What I was shown was to pull the glue up to the edge with one way strokes perpendicular to the edge. The shop teacher brushed parallel to the edge with a wet brush... the result was a complete mess and he did in fact gum up the bearing on the router bit. Any time saved by using the long parallel strokes was lost in the clean up and failed bearings.
Now... I see short knapped rollers and sprayers being used in commercial shops, like the one a relative of mine operates. He has a room where the glue is sprayed that has filters and an exhaust system. Not a brush or squeegee in sight! He has an automated machine to apply edge banding.
I don't imagine BBR or Ironhorse taking the neat route.