In the past, I have always applied edge treatments like round-overs or chamfers before I do any finishing prep like sanding or scraping. ...wonder if perhaps I am doing this in the wrong order.
No this is the right order. Sanding is as you say part of the finishing process, smoothing the wood in preparation for a finish to go on. So it should as much as possible always be done after any shaping steps, otherwise you're wasting effort smoothing some wood that will subsequently be removed.
The problem is with the highlighted portion below.
The router bits leave what looks like a nicely finished edge. I then go back over these with 80grit or 120grit to start, which usually initially makes them appear worse than before.
Any time a process in woodworking makes a surface look less good than it did you should be thinking "Am I doing the right thing here?"
80 grit is not a smoothing/finishing grit, it's a grit coarse enough to be used for material removal, i.e. a shaping operation.
If your routing is leaving a pretty good surface you should be sanding them minimally, just enough to even up the surface texture so that it accepts finish approximately the same as the flat portions of the workpiece. Some people don't even sand their routed profiles, but it's rare you can get away with that.
The goal with all final shaping steps should be that only minimal sanding is required after, and although people do still commonly work up through the grits in two or three steps finish sanding can often be accomplished using just a single grit. The grit to use for this can be as coarse as 120 occasionally, but 150 is the usual lower limit and 220 a common upper limit, with 180 being in the sweet spot for many woodworkers.
