Dov,
PitStop has had the ability to add a bleed box, you can then move or stretch elements to extend into the bleed.
In the latest version they have an improved action list to add a bleed box and extend both vector and raster elements into the bleed.
The raster images are extended by mirroring the edge, yes, it looks a bit weird, but then again, it is the bleed which will be trimmed off.
We all know your feelings about opening PDF in Illustrator, and we agree that Illustrator is NOT a PDF editor.
So isn't it better to use PitStop to adjust elements in a PDF rather than using Illustrator or Photoshop?
Now it seems like the original poster will continue to use AI & PS for editing PDFs when a better tool is available.
Here is an Enfocus video showing the add bleed function: Detect missing bleed and fix it with PitStop Pro 13 - YouTube
PitStop can do many other things with PDFs, not just bleeds.