> When you store a value into memory, the register renamer does nothing, because you do not attempt to modify any register.

you are of course correct about everything. But the extreme pendant in me can't avoid pointing out that there are in fact a few mainstream CPUs[1] that can rename memory to physical registers, at least in some cases. This is done explicitly to mitigate the cost of spilling. edit: this is different from the store-forwarding optimization you mentioned.

[1] Ryzen for example: https://www.agner.org/forum/viewtopic.php?t=41