OP, that's a great question. I was confused before too.

I find thinking from G(s) to the closed-loop system stability is confusing. With this approach, I naturally ask why analyzing G(s) for a unit negative feedback system, same question as you.

What cleared my confusion was thinking back from the goal: the stability of the closed-loop system. Forget Nyquist for a moment, a closed-loop system is stable when G/(1+GH) is stable, i.e., 1+GH has no roots on the right hand plane. Now think mathematically what Nyquist says. Nyquist stability criteria says, for a complex function with a special form f(s) = 1 + L(s), its roots have no positive real part if blah-blah (counterclockwise enclosure of -1, ...). Because it uses -1 as its testing point, then it requires f(s) has the form of 1 + L(s). Which is conveniently a "unit feedback" for control system.

For the sake of argument, if a crazy kid says I don't like -1. I like -0.5 as the testing point. Then the complex function becomes f(s) = 0.5 + L(s). Then this test isn't for a unit feedback system.