Either you have a way to address an offset into the stack, or you don't. If you do, the stack is conceptually equivalent to a register file where you can optionally swap out the entire content in one go (by changing the stack pointer).
If you don't, then registers will in most designs offer you a strict superset of operations: I've yet to see any register-based machines where you can't load things into registers, do the operation and push them back on the stack. In fact, on most machines that are not purist RISC designs, you will tend to be able to do many/most operations with one of the operands in memory.
That superset comes at a complexity cost in some areas, sure, but being able to address more different data items without having to manipulate the stack also turns out to be very handy. If it wasn't, we'd just stick to mostly manipulating the stack even in register-based designs.
EDIT: In fact, you'll often see "naive" first code generators for register-based machines heavily rely on stack manipulation for expression evaluation because it massively reduces the need for register allocation complexity.