register-register movs are needed because compilers are not the world's smartest code generators, and we don't have an infinite number of registers. The biggest problem arises when some calling convention expects certain things to be in certain registers - unless you do some pretty extreme long term planning in your register allocation code in the compiler, you're pretty likely to eventually wind up with some piece of information in a register that you need to move to another register before making or returning from a call.

Of course, modern processors don't really care about register-register moves because they have hundreds of hidden intermediate registers and register renaming which makes those types of moves "free." (e.g. Haswell has 168 GP registers, only 16 of which are exposed through the AMD64 ISA.)