Certainly you could build a cab with just a router. You could also do it with just hand tools. Hell, you could probably build the pyramids with a hammer and chisel (and some slave labor). But when placing a value on time, labor (and frustration avoidance) certain tools are a must.
I don't absolutely need a drill press, but I had to drill well over 300 holes for my cab and a drill press is a god-send for that. In fact there are things I drilled that wouldn't be possible with a hand-drill.
I don't absolutely need a table saw, but to cut 3 panels EXACTLY the same size, each with a 17 degree bevel, required multiple attempts with a circular saw, but just a single pass on a table. In fact every cut I made with my circular saw, I re-did with a router, jig-saw or table saw, so in effect, I never used my circular saw for my cab.
I don't absolutely need a dremel, but to cut 4 piano hinges and metal brackets down to usable size was much easier, quicker & cleaner than using a hack-saw.