Hindley-Milner system with let expansion
I think something somewhat trivial is happening here, that is if the let binding is not used, then you can let bind an ill-typed term, e.g. if:

$t =\ $let x = (if 3 then 4 else false) in 5

then $t^* =\ $ 5 which is well-typed in $\mathrm{HM}$, even though t is not. If every let-binding is used, though, then this pathology can't happen.