I've worked as a mechanical engineer and done metal part design. That doesn't really sound like a problem caused by computer aided drafting so much as just human laziness or incompetence.
Using *mechanical* design software like SolidWorks or ProE or Catia that stuff is pretty simple. AutoCAD, which I haven't used extensively but God knows I've tried and given up on it a few times, is basically just a shitty drawing program.* It's really old and well established though and it's locked in as the standard for civil engineering just because it's what everyone else uses, like MS Office or Photoshop (except those don't suck as much). Note I said civil engineering; designing mechanical parts in AutoCAD is really old school and dumb IMO. But SolidWorks and the others let you see your part in 3D, and assemble it together with other parts to see how they fit, and with plug-ins or separate software you can also model the stress and strain (bending, stretching, etc.) when it's put under weight at a given point. You can also do all those calculations by hand/in a separate math or modeling program just as you always could. And typically prototypes are manufactured and physically tested. AutoCAD's not supposed to replace the physical testing; it just replaces the pencil and paper. If people are skipping testing that's just good old corner cutting.
*It does have a lot of power user features and it may actually have the stuff I'm saying it lacks, just buried behind an impenetrable layer of hard-to-useness and suckiness. I haven't invested the time to become an AutoCAD expert because it's always been easier to just use something else.
I'm pretty sure AutoCAD can do all that stuff. I know it does the 3D stuff - the part I referenced was designed in a 3D model in AutoCAD. The problem is all the layering you have to do so see everything fitting together - updating every single diagram to show the new parts. Now, may be AutoCAD and others make that easier than I realize (I'm no expert at them); and I really don't know why they didn't figure out that the parts wouldn't fit (in this case, the one end of the piece made and the piece it attached to couldn't be physically accessed because other stuff interfered on one side but not the other, or at least the other wasn't as bad).
I wouldn't be surprised by a "shitty drawing" though as the one ME in charge was not liked very well by the technicians since this was not an abnormal thing - everything being in the drawing, but not necessarily being obvious how to do it, even when multiple detailed diagrams were provided. But that's a different thing entirely.
When talking about cars - I'm referring to things like you have to take off 10 parts that requires 8 hours of labor to do in order to change 1 part that takes only 30 minutes of labor; when providing a little more space could have enabled one to change that part in 30 minutes entirely without to disassemble a chunk of the vehicle to get to it.