Because with 10 gauge wire, you could have about two loops around the pickup. Resistance is not important in a pickup coil. What IS important is the number of loops.
Here's how a pickup actually works: The magnets make a fixed magnetic field. The strings vibrate through said magnetic field, which gives the strings a changing magnetic flux and induces a current IN THE STRINGS. That induced current in the strings then generates a changing magnetic field on top of the static one. The coils of the pickup have a current induced by the changing magnetic flux generated by the induced current in the strings, and that current is what goes to the amp.
The current induced in a coil is proportional to the number of loops in the coil and the magnetic flux in the interior of the coil and has NOTHING to do with the resistance of the wire. This comes from Stokes' law, which in this case says that the line integral of the curl of the magnetic flux around the coil is proportional to the integral of magnetic flux through the coil itself. Adding extra loops means the flux isn't enclosed once, it's enclosed as many times as there are loops - making five thousand loops instead of one is the same as making the flux five thousand times stronger with a single loop.
The magnetic flux created by the induced currents in the moving guitar string are TINY. A single loop of wire enclosing that flux would generate such a tiny induced voltage (on the order of a tenth of a millivolt, perhaps). Moving to, say, five thousand loops multiplies that induced voltage by five thousand, up to something on the order of a volt, a usable voltage that can be transmitted down a guitar cord to an amplifier and amplified without too much background noise.
You need a TON of loops to have a coil sensitive enough to pick up the tiny magnetic field flux generated by the moving strings - so you need to use thousands of wraps of tiny wire. Having one loop of really thick wire simply does not work.