Why do you plug a DI box into the microphone input?
As with many things, the origin of this convention is historical. Briefly, the reason why a DI box is necessary to connect an electric or bass guitar directly to the mixing console is that the pickups are strong in voltage but very weak in current. The microphone input of a mixing console requires a good helping of current; the line input doesn't need so much current, but still more than the pickups can supply.
However, the mic input requires only a small voltage. So if some of the relatively high voltage from the pickups could be exchanged for current, then it would drive a mic input perfectly well.
And the simple solution is a transformer. This simple device consists of two coils of wire wrapped around the same metal core. According to the number of turns on each coil, a transformer can step up voltage while stepping down current, or vice versa.
So the original DI boxes were just transformers, and they had to connect to the mic input to work properly. Transformers don't need any power supply to work, which can be seen as a convenience.
Modern DI boxes use electronics rather than transformers, mainly because good transformers are expensive. And since they were intended to replace trasnformer DI boxes in the marketplace, they had to function in the same way. Hence an electronic DI also connects to the mic input.
But what about those batteries - isn't it a hassle to keep replacing them all the time? Well no, because the DI box can take advantage of the phantom power provided by the mic input. No batteries necessary, although you can usually fit them if you need to use the DI in some other way.
So although an electronic DI box could be designed to connect to the line input, it wouldn't receive the benefit of phantom power. Connecting to the mic input works just fine and everyone wins all round.
Except battery manufacturers.