Inspired by @Joey Pongallo, I started researching pre-made relay boards instead of creating my own MOSFET-based switches. It turns out that most of these boards are made for the Arduino eco system and thus require 5V levels to switch. Some people report that their relay boards work with 3.3V logic levels, but that typically requires a separate 5V supply for the relay coils. Even then, the logic levels may be too low to drive the switching transistor to saturation. This results in a very hot transistor.
During my research, I stumbled upon cheap Arduino clones. These boards provide 5V logic levels and the entire Arduino eco system is built around them, except for some oddball Arduinos running on 3.3V.
I've ordered an Arduino Nano V3 clone and a 4-relay module for 13€ total including shipping. This is a much better option than spending another several hours designing and soldering the MOSFET-based switch.
As a plus, the board can run on supply voltages up to 20V as it has a built-in voltage regulator. I could - theoretically - run the relay coils off that regulator. I will have to see about that once the board arrives and I can determine the particular type of regulator used. I would have preferred a 12V relay, but these are quite rare and also more expensive.
If the built-in regulator can't supply enough current (70mA per coil), I still have my LM2596 based converter.