You couldn't do random, but with a predictable performance hit to memory, cache and write-line use stack addresses COULD be isolated for a program, for a library, etc.
It'd be expensive though; every context switch would require it's own stack and pushing / restoring one more register. There's GOOD reason programs don't work that way and are supposed to not rely on values outside of properly initialized (and not later clobbered) memory.
Why would a specialized instruction be necessary? 'the stack' is stored in memory just like everything else.
Expensive is the (very slow for modern CPUs) operation of _writing_ that change in value out to memory at it's distant and slow speed compared to that which the CPU operates at, as well as the overhead of synchronizing that write to any other caches of those memory locations.
Maybe you're thinking of the trick of a band new page of memory mapped memory that is 'zeroed' but is in reality just a special 'all zeros' page in the virtual to physical memory lookup table? Those still need to be zeroed by real writes at some point, if they're ever used.
It should be efficient though, that's the point. Specialized hardware or instructions should be able to zero the stack in a single cycle, instead it's much more expensive. Of course the problem with this is it could be used to hide things just as easily, making it impossible to reverse engineer an unknown exploit.