Agreed, the beneficial owner has the ultimate responsibility as...

Agreed, the beneficial owner has the ultimate responsibility as to whether to make securities available to the programme.
In the case of a fund, it's the issuer (Blackrock in your example).
In the case of a separately managed account (let's say a Hesta Active Aust Equities mandate) then it's their call in their agreement with the custodian

Knowing the capability of some (not all) people involved in the direct management of assets at many industry funds, it would not surprise me if they'd never heard of a SBL programme meant or that their underlying assets were being put to use. But that's another topic altogether.