The only other thing I can think of is that you did want $closure in ... to reference to same variable.

Exactly.

With this syntax:

{
    my $closure;
    sub x1{ ... }
    sub x2{ ... }
    sub x3{ ... }
    ...
}
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I can share the closure between as many subroutines as I need.

With embedded state, the closure's scope is restricted to the one subroutine.

I could do:

{
    state $closure;
    sub x1{ ... }
    sub x2{ ... }
    sub x3{ ... }
    ...
}
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But then state has no advantage over the simple closure.

I did start to use state when it first appeared (there are probably several examples here somewhere), but after the third or fourth time of having to revert it back to a standard closure as needs changed, I simply stopped using it.