The main issue is mission assurance. Using the stack or the heap means your variables aren't always at the same memory address. This can be bad if a particular memory cell has failed. If every variable has a fixed address, and one of those addresses goes bad, a patch can be loaded to move that address and the mission can continue.
Your mention of STL makes it sound like you're talking about C++. But I don't know of any C++ compiler that lets you completely avoid use of the stack, even if you disable the usual suspects (RTTI and exceptions). Sure, you'd have to avoid local variables, defined within a function's body (or at block scope), but that's nowhere near enough.
* The compiler would need to statically allocate space for every function's parameters and return address. That's actually how early compilers did work, but today it would be inefficient because there are surely so many functions defined in a program's binary compared to the number being executed at any given time. (Edit: I suppose you already need the actual code for those functions, so maybe allocating room for their parameters is not so bad.)
* It would also mean that recursion would not work, even mutual recursion (so you'd need runtime checks because this would be hard to detect at compile/link time), although I suspect this is less of a problem than it sounds, but I'm not aware of a C++ compiler that supports it.
* You'd also need to avoid creating any temporary variables at all e.g. y = a + b + c would not be allowed if a,b,c are non-trivial types. (y = a + b would be OK because the temporary could be constructed directly into y's footprint, or stored temporarily in the return space of the relevant operator+(), which again would be statically allocated).
Is that really what you meant? I suspect not, but without all that your point about avoiding the stack doesn't make any sense.
In most other surviving CPU ISAs the return address is saved in a register and it is easy to arrange in a compiler to use only procedure arguments that are passed in registers, the only price being paid for this being a reasonable upper limit for the number of parameters of a function, e.g. 12 or 24, depending on the number of general-purpose registers (e.g. 16 or 32). For the very rare case when a programmer would want more parameters, some of them should be grouped into a structure.
With this convention, which normally should not be a problem, there is no need for a call stack. There can be software managed stacks, which can be used even for implementing recursion, when that is desired.
The use of static memory for passing function arguments was necessary only in the very early computers, which were starved in registers.
1. Where do you save the current value of the return address register before calling a function?
2. When parameters are "grouped into a structure" and the structure is passed as an argument to a function, where do you store that structure?