I have a circuit that should be turned on whenever the car stereo is turned on. It will draw ~300mA. My plan is to connect my circuit to the wire harness behind the stereo as it has +12v, ground and ignition swith-on signal available. Is there a reason why I shouldn't connect it directly to the car stereo wire harness as shown in the picture below?
That's exactly how most circuits in a car are wired. Parallel circuits. The stereo power line is normally wired to a 10A or higher fuse and 300 mA won't be a problem.
Without qualifications, this typically works. As a note, the common circuitry implementations for cars seems to be: the unswitched power circuits provide the bigger, continuous power supply; the smaller switched, ignition-on, are controls circuits. Yes, they are both electrical power, but each has different intentions. Smaller control wire, connectors, labor, space and options can be done this way. E.g. several things may control the headlights, but they are powered by the same power circuit. You may also want to note the circuit protection size you are tapping into as it may not protect where the device needs it. You also may need to ground away from high current grounds (ground loop). (If you're possibly adding a steering wheel module.)
