I work with lot of architects and design school graduates and students that still work without computers. There is really nothing wrong with doing it this way except that it neccesitates sometimes having somebody else make your drawings digital. Nothing like having to convert a carboard mockup to 3D model.
The way you do this is by taking a sheet of paper and start drawing. Then after you got a rough idea you draw more just to be sure. Then you draw a clean drawing on top of that. But by not having a CAD also frees you, you can bend sheetmetals out of cardboard, you can sculpt stuff out of modeling clay, or sand out of a block of polyurethane.
You might spot at this point that the downside is that any small change and youll be redoing things. A lot. I used to work with a older person who had been a draftsman who said that he basically drew one drawing a day, only to start over next day for a trivial change.
Now while this is a bit bad for job productivity, theres also some advantages of working on paper. The problem with MCAD software like solidworks is that its designed for the detail work stage. The problem that raises with this is that it railroads you to thinking certain ways. Having paper sketches is a way to combat this thing. But it has advantages too, it lowers your cost, having a extra draftsperson is expensive.
It is true that the CAD application has in a way killed the drafing job. But please note, it is culturally dependent, in some parts of the world the engineers did their own drafting even before the advent of CAD applications. Today even if companies want to have somebody to do just the technical drawings, they can not easily find draftsmen (this is to my understanding much more a european problem). So if you want to have somebody you have to have a junior person doing this. This isnt ideal as a senior person would take care of other details too.
Which brings us to the second person that the modern CAD application also killed off many of the modelmaker positions. So it used to be that you submitted a functional drawing to a modelmaker and they would just add drafts etc etc. Also you have a lot less people in general these days. So the CAD application really kills of 2-8 persons in the loop depending on how far back you look back.
The problem with having a extra person in the loop is that the change I want takes one day, while if im modeling myself it takes often less than a hour. But then i have more chances for screwups.
To answer your final question enfineers had mostly the same duties as they have today, using the cad is just a extra tool layer on top. Not all mechanical engineers draw in CAD.