Why should anyone need to mess with the A20 line and the keyboard controller to simply reset a computer?

Well, you do need to turn on A20 so it will jump to right address, but if you are in protected mode it should already be on.

Isn't that kludgy? I mean, using a keyboard controller (in effect, a second computer) to reset the one you can't?

The reason was that IBM bought the rest of the chips "as they were" from Intel, so all the logic that they made themselves ended in the in-house keyboard controller. x86 had an external interrupt line for reset, and an io pin in the keyboard controller was tied to this. That was a reasonable way to get true hard reset back then.

And what I meant by related was that another pin in the same keyboard controller was tied to the A20 gate, so it was toggled in a similar way.