You have it backwards. Modern compilers don't use SSA because it's "simpler", we use it because it enables very fast data-flow optimizations (constant prop, CSE, register allocation, etc.) that would otherwise require a lot of state. It doesn't "pretend state doesn't exist", it's actually exactly what makes it possible/practical for the compiler to handle changes in state.

As some evidence to the second point: Haskell is a language that does enforce immutability, but it's compiler, GHC, does not use SSA for main IR -- it uses a "spineless tagless g-machine" graph representation that does, in fact, rely on that immutability. SSA only happens later once it's lowered to a mutating form. If your variables aren't mutated, then you don't even need to transform them to SSA!

Of course, you're welcome to try something else, people certainly have -- take a look at how V8's move to Sea-of-Nodes has gone for them.