They don't need to use mortgage financing, they can borrow the money through other methods. One consequence of Fed Reserve rate setting in combination with a regulated mortgage market is that private equity firms can borrow at lower rates than individual borrowers. I wouldn't be surprised to learn that this dynamic has been a major contributor to housing prices rising so quickly.
BlackRock buys stakes in investment firms that directly purchase real estate (some specifically SFH).
Those companies issue corporate debt (bonds) to buy houses.
The yield they need to offer goes down when the Fed manipulates bond prices by gobbling up non-corporate debt (US Treasuries & MBS).
BlackRock & other firms aren't taking on debt to invest in REITs - they're just directing more of their portofolio to REITs when The Fed drives down yields.