> modern CPUs pipeline this shit

Quite. In practice, a high-performance processor today, either stack or register, will internally have little resemblance to the presented instruction set. The stack will become a huge array of registers that are the inputs and outputs to various computation units, and they will be tracked and renamed as necessary to align with the various slots in the stack, rather than actually moving things around on the stack. Just as the integer register file on a register machine, becomes a huge array of registers that are renamed as necessary. Any difference between them, in terms of hardware implementation, has largely disappeared over the last couple decades.