They aren't qualified, because the REIT is a special kind of company, and not double-taxed like a regular corporation (as you mention). The income from the real estate holdings gets passed straight through, similar to how a mutual fund passes any interest or capital gains it earns straight through to shareholders.
To avoid the double tax the REIT has to agree to distribute 90% of its income to shareholders, rather than keeping it inside the company. That ensures that while the taxation is only "single," it doesn't get deferred indefinitely into the future. For a normal corporation, paying a dividend is optional and can be put off forever if desired.