I've reconnected several of these (C1, U1 GND) more than once, even redrawing entire wire segments, but I still get errors about unconnected pins. Is there something I'm doing incorrectly?
The blue lines are busses. They do not connect signals without a specific naming convention (http://docs.kicad-pcb.org/5.0.0/en/getting_started_in_kicad.html#bus-connections-in-kicad).
Use wires (green lines) instead.
I think those blue lines are busses, not wires. Busses are usually just graphic items indicating a bunch of signals. Each wire connected to the bus must have a signal name - the connectivity is via the signal names.
If you replace the bus lines with wires, and remove the 45 degree buss connectors, things should work.
I would only use bus lines for a data or address bus - definitely not for Ground and Vcc, or for a single signal.