Is it likely that we'd get enough usage out of an airship to invest in the Workshop and parts?
Yes. If your end-game goal is crafting and gathering, you're going to want airships to fund and supply the amount of materia you're going to blow up.

With only four (and only two who play a lot) is all of it going to be so expensive that we'll feel overwhelmed by the costs or the farming for materials? Will progress be really slow?
I founded a FC and built 4 end-game airships (eventually going to do a second FC with 4 more airships) with just 2 players. Some parts will seem overwhelming, but the investment pays off. My suggestion is to farm and avoid the MB when you can.

Will the four of us be able to clear enough content in the Diadem to make it worth going? We aren't interested in finding four or more other people to go with and obviously we're not looking to do Hard mode.
As far as gathering, 4 people in is more than enough. I do it with 2 (sometimes solo with Titan-egi to get the currants if I can get a friend to help me queue in when my FC mate is offline). I'm not sure if 4-manning the mobs in Diadem is viable yet, but I haven't tried.