If I were to put the parts of the guitar in order of importance when it comes to how much they each influence sound, I would put the pickups at the top of the list. The pickups, more than any other component of an electric guitar, have the power to shape the sound. They drastically influence the character of an instrument, and they are extremely important, along with the capacitors and the rest of the electronics inside an electric guitar.
But other things are important, too, such as the tonewoods chosen to build the instrument. When I say this, some guitar players immediately take it as some kind of challenge against pickups and electronics, but that’s not the point. Yes, pickups may matter more than anything, but other things matter, too. The wood. The nut. The bridge. The way the neck is attached. The body cavities inside the guitar. Even the paint.
An electric guitar isn’t some artless electrical thing thing like a toaster or microwave oven - it's a complex machine, and someone has put a great deal of thought into the design. Yes, the electronic technology is important, but so is the craftsmanship. Pickups may matter the most, but I think everything matters. To whittle the factors that influence the sound of an electric guitar down to the pickups alone is extremely over simplistic.