If I purchased the Kitfox one section at a time, could I actually do any work
Absolutely YES.

I'm on my 2nd Kitfox purchased in partial kits only. My model 4 was completed and flying in 15 months buying only the fuselage kit from the factory.

I just finished my Super Sport 7 fuselage kit through covering in a little over 3 months. It sits awaiting paint booth space to open up while I do a hose refresh on my model 4 Rotax engine.
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My 2 cents- if you can't swing the entire kit purchase but really want to get building then the best bet is to get the fuselage kit first. You can complete it to covering with all included materials. Obviously the tail feathers are included also.
The fuselage represents somewhere around 70 % of your entire kit build time if wiring , landing gear, engine mounting, cowling fitting are considered.

My only concern there is that if you took each component kit to completion through covering, you would want to be very thorough in your planning, i.e. running wires and tubing - example - nav and anti-collision lights and pitot and static port.
I have only one set of wires running to the tail and that is the trim system wiring that comes with the fuselage kit. I do have my comm radio antenna mounted in the vertical stabilizer so I have a comm co-ax cable run to it. Most other wiring is in the cockpit or behind the panel except wing lights, pitot tubing ect.