The stochastic parrot is much more grounded in the physical reality of how the models are structured, trained, and produce outputs than the "it's like people" argument. Statistical models are much better understood than biological systems. That said, your penultimate statement is true:

It's entirely possible that every trait you consider human can naturally emerge from a dumb statistical model, and indeed certain processes are remarkably similar as the models get bigger, smarter, and have better training data.

I just don't think it's likely given the biological complexity we are washing over with the statistical model.