Nested rules and automatic renaming of variables

Question

I understand why $ happens here:

ClearAll[foo]
foo[a_] := (x_ :> a x)

foo[1]

x$_ :> 1 x$

For future readers, see: Enforcing correct variable bindings and avoiding renamings for conflicting variables in nested scoping constructs

But I don't understand this one:

ClearAll[foo]
foo[a_] := (x_ :> a x) -> a

foo[1]

(x$_ :> 1 x) -> 1

Only one x is renamed. Should I expect this or is this a bug?

Somehow I feel this one is related:

Bug with iterated With: Referring to function arguments invalidates previous definitions

p.s. here is a oneliner:

Key["a"] /. Key[name_] :> ((whatever_ :> whatever) -> name)

(whatever$_ :> whatever) -> "a"

The real use case is something along

    Key["a"] /. Key[name_] :> (( 1 /. whatever_ :> whatever) -> name)

Where I'd expect 1 -> "a" instead of whatever -> "a"