Institutions lend shares they own for shorting all the time,...

Institutions lend shares they own for shorting all the time, just to gain the tiny fee for doing so. Let's say the institution is a Superannuation  Fund. They hold NCZ for the long term in their Default Fund because it is an ASX300 stock. They don't care if NCZ's price goes down when they lend for shorting. because the Member loses, not the fund. The Fund makes its money from the fees it charges the Members. As long as you are a member, you are paying fees. The members take the price risk.