If I understand you correctly, the cut will be out of sight and one wall of the open box will be acting as a fence, so either the box or the router will have to move, then lock down, at which point you will run into all the problems described. 2x lumber is notoriously crooked, warped, twisted, cracked, so an edge guide is a much more practical solution. But even then, a "crooked" piece will still be off.
It sounds like you're going to run a lot of pieces at one time, and honestly I'd hate to try to run, say, 50 pieces through what you described. It would certainly be far faster to use an edge guide on construction grade lumber, using a 3hp router would speed things up some, but heating and resin buildup on that many pieces would probably require cleaning the bit several times, plus a short cool off period every couple of boards.
What I would want for a large number of pieces would be a flat, well-supported surface with a couple of pegs to hold the board solidly in place as you rout it. If you are edging extra long pieces, you'd want to clamp one edge so it doesn't shift when you get to the ends.
Love questions like this, makes the brain and imagination work.
The more I do, the less I accomplish.